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== 9.4#h(0.6em)Multiply Square Roots

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Before you get started, take this readiness quiz.

Simplify: #math.equation(block: false, alt: "open parenthesis 3 u close parenthesis open parenthesis 8 v close parenthesis")[$( 3 u ) ( 8 v )$]. #linebreak() 
If you missed this problem, review .

#solutionbox[
#math.equation(block: true, alt: "24 u v")[$24 u v$]
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Simplify: #math.equation(block: false, alt: "6 open parenthesis 12 minus 7 n close parenthesis")[$6 ( 12 − 7 n )$]. #linebreak() 
If you missed this problem, review .

#solutionbox[
#math.equation(block: true, alt: "72 minus 42 n")[$72 − 42 n$]
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Simplify: #math.equation(block: false, alt: "open parenthesis 2 plus a close parenthesis open parenthesis 4 minus a close parenthesis")[$( 2 + a ) ( 4 − a )$]. #linebreak() 
If you missed this problem, review .

#solutionbox[
#math.equation(block: true, alt: "8 plus 2 a plus a squared")[$8 + 2 a + a^(2)$]
]
]


=== Multiply Square Roots

We have used the Product Property of Square Roots to simplify square roots by removing the perfect square factors. The Product Property of Square Roots says

#math.equation(block: true, alt: "the square root of a b equals the square root of a times the square root of b")[$sqrt(a b) = sqrt(a) · sqrt(b)$]We can use the Product Property of Square Roots ‘in reverse’ to multiply square roots.

#math.equation(block: true, alt: "the square root of a times the square root of b equals the square root of a b")[$sqrt(a) · sqrt(b) = sqrt(a b)$]Remember, we assume all variables are greater than or equal to zero.

We will rewrite the Product Property of Square Roots so we see both ways together.

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
#emph[Product Property of Square Roots]

If #emph[a], #emph[b] are nonnegative real numbers, then

#math.equation(block: true, alt: "the square root of a b equals the square root of a times the square root of b and the square root of a times the square root of b equals the square root of a b")[$sqrt(a b) = sqrt(a) · sqrt(b) #h(1em) "and" #h(1em) sqrt(a) · sqrt(b) = sqrt(a b)$]
]

So we can multiply #math.equation(block: false, alt: "the square root of 3 times the square root of 5")[$sqrt(3) · sqrt(5)$] in this way:

#math.equation(block: true, alt: "the square root of 3 times the square root of 5; the square root of 3 times 5; the square root of 15")[$sqrt(3) · sqrt(5) \ sqrt(3 · 5) \ sqrt(15)$]Sometimes the product gives us a perfect square:

#math.equation(block: true, alt: "the square root of 2 times the square root of 8; the square root of 2 times 8; the square root of 16; 4")[$sqrt(2) · sqrt(8) \ sqrt(2 · 8) \ sqrt(16) \ 4$]Even when the product is not a perfect square, we must look for perfect-square factors and simplify the radical whenever possible.

Multiplying radicals with coefficients is much like multiplying variables with coefficients. To multiply #math.equation(block: false, alt: "4 x times 3 y")[$4 x · 3 y$] we multiply the coefficients together and then the variables. The result is #math.equation(block: false, alt: "12 x y")[$12 x y$]. Keep this in mind as you do these examples.

#examplebox("Example 1")[][
Simplify: ⓐ #math.equation(block: false, alt: "the square root of 2 times the square root of 6")[$sqrt(2) · sqrt(6)$] ⓑ #math.equation(block: false, alt: "open parenthesis 4 the square root of 3 close parenthesis open parenthesis 2 the square root of 12 close parenthesis")[$( 4 sqrt(3) ) ( 2 sqrt(12) )$].

#solutionbox[
ⓐ

#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([], [#math.equation(block: false, alt: "the square root of 2 times the square root of 6")[$sqrt(2) · sqrt(6)$]]),
  [Multiply using the Product Property.], [#math.equation(block: false, alt: "the square root of 12")[$sqrt(12)$]],
  [Simplify the radical.], [#math.equation(block: false, alt: "the square root of 4 times the square root of 3")[$sqrt(4) · sqrt(3)$]],
  [Simplify.], [#math.equation(block: false, alt: "2 the square root of 3")[$2 sqrt(3)$]],
))

ⓑ

#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([], [#math.equation(block: false, alt: "open parenthesis 4 the square root of 3 close parenthesis open parenthesis 2 the square root of 12 close parenthesis")[$( 4 sqrt(3) ) ( 2 sqrt(12) )$]]),
  [Multiply using the Product Property.], [#math.equation(block: false, alt: "8 the square root of 36")[$8 sqrt(36)$]],
  [Simplify the radical.], [#math.equation(block: false, alt: "8 times 6")[$8 · 6$]],
  [Simplify.], [#math.equation(block: false, alt: "48")[$48$]],
))

Notice that in (b) we multiplied the coefficients and multiplied the radicals. Also, we did not simplify #math.equation(block: false, alt: "the square root of 12")[$sqrt(12)$]. We waited to get the product and then simplified.
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Simplify: ⓐ #math.equation(block: false, alt: "the square root of 3 times the square root of 6")[$sqrt(3) · sqrt(6)$] ⓑ #math.equation(block: false, alt: "open parenthesis 2 the square root of 6 close parenthesis open parenthesis 3 the square root of 12 close parenthesis")[$( 2 sqrt(6) ) ( 3 sqrt(12) )$].

#solutionbox[
ⓐ #math.equation(block: false, alt: "3 the square root of 2")[$3 sqrt(2)$] ⓑ #math.equation(block: false, alt: "36 the square root of 2")[$36 sqrt(2)$]
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Simplify: ⓐ #math.equation(block: false, alt: "the square root of 5 times the square root of 10")[$sqrt(5) · sqrt(10)$] ⓑ #math.equation(block: false, alt: "open parenthesis 6 the square root of 3 close parenthesis open parenthesis 5 the square root of 6 close parenthesis")[$( 6 sqrt(3) ) ( 5 sqrt(6) )$].

#solutionbox[
ⓐ #math.equation(block: false, alt: "5 the square root of 2")[$5 sqrt(2)$] ⓑ #math.equation(block: false, alt: "90 the square root of 2")[$90 sqrt(2)$]
]
]

#examplebox("Example 2")[][
Simplify: #math.equation(block: false, alt: "open parenthesis 6 the square root of 2 close parenthesis open parenthesis 3 the square root of 10 close parenthesis")[$( 6 sqrt(2) ) ( 3 sqrt(10) )$].

#solutionbox[
#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([], [#math.equation(block: false, alt: "open parenthesis 6 the square root of 2 close parenthesis open parenthesis 3 the square root of 10 close parenthesis")[$( 6 sqrt(2) ) ( 3 sqrt(10) )$]]),
  [Multiply using the Product Property.], [#math.equation(block: false, alt: "18 the square root of 20")[$18 sqrt(20)$]],
  [Simplify the radical.], [#math.equation(block: false, alt: "18 the square root of 4 times the square root of 5")[$18 sqrt(4) · sqrt(5)$]],
  [Simplify.], [#math.equation(block: false, alt: "18 times 2 times the square root of 5")[$18 · 2 · sqrt(5)$]],
  [], [#math.equation(block: false, alt: "36 the square root of 5")[$36 sqrt(5)$]],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Simplify: #math.equation(block: false, alt: "open parenthesis 3 the square root of 2 close parenthesis open parenthesis 2 the square root of 30 close parenthesis")[$( 3 sqrt(2) ) ( 2 sqrt(30) )$].

#solutionbox[
#math.equation(block: true, alt: "12 the square root of 15")[$12 sqrt(15)$]
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Simplify: #math.equation(block: false, alt: "open parenthesis 3 the square root of 3 close parenthesis open parenthesis 3 the square root of 6 close parenthesis")[$( 3 sqrt(3) ) ( 3 sqrt(6) )$].

#solutionbox[
#math.equation(block: true, alt: "27 the square root of 2")[$27 sqrt(2)$]
]
]

When we have to multiply square roots, we first find the product and then remove any perfect square factors.

#examplebox("Example 3")[][
Simplify: ⓐ #math.equation(block: false, alt: "open parenthesis the square root of 8 x cubed close parenthesis open parenthesis the square root of 3 x close parenthesis")[$( sqrt(8 x^(3)) ) ( sqrt(3 x) )$] ⓑ #math.equation(block: false, alt: "open parenthesis the square root of 20 y squared close parenthesis open parenthesis the square root of 5 y cubed close parenthesis")[$( sqrt(20 y^(2)) ) ( sqrt(5 y^(3)) )$].

#solutionbox[
ⓐ  #linebreak()

#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([], [#math.equation(block: false, alt: "open parenthesis the square root of 8 x cubed close parenthesis open parenthesis the square root of 3 x close parenthesis")[$( sqrt(8 x^(3)) ) ( sqrt(3 x) )$]]),
  [Multiply using the Product Property.], [#math.equation(block: false, alt: "the square root of 24 x to the power 4")[$sqrt(24 x^(4))$]],
  [Simplify the radical.], [#math.equation(block: false, alt: "the square root of 4 x to the power 4 times the square root of 6")[$sqrt(4 x^(4)) · sqrt(6)$]],
  [Simplify.], [#math.equation(block: false, alt: "2 x squared the square root of 6")[$2 x^(2) sqrt(6)$]],
))

ⓑ  #linebreak()

#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([], [#math.equation(block: false, alt: "open parenthesis the square root of 20 y squared close parenthesis open parenthesis the square root of 5 y cubed close parenthesis")[$( sqrt(20 y^(2)) ) ( sqrt(5 y^(3)) )$]]),
  [Multiply using the Product Property.], [#math.equation(block: false, alt: "the square root of 100 y to the power 5")[$sqrt(100 y^(5))$]],
  [Simplify the radical.], [#math.equation(block: false, alt: "10 y squared the square root of y")[$10 y^(2) sqrt(y)$]],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Simplify: ⓐ #math.equation(block: false, alt: "open parenthesis the square root of 6 x cubed close parenthesis open parenthesis the square root of 3 x close parenthesis")[$( sqrt(6 x^(3)) ) ( sqrt(3 x) )$] ⓑ #math.equation(block: false, alt: "open parenthesis the square root of 2 y cubed close parenthesis open parenthesis the square root of 50 y squared close parenthesis")[$( sqrt(2 y^(3)) ) ( sqrt(50 y^(2)) )$].

#solutionbox[
ⓐ #math.equation(block: false, alt: "3 x squared the square root of 2")[$3 x^(2) sqrt(2)$] ⓑ #math.equation(block: false, alt: "10 y squared the square root of y")[$10 y^(2) sqrt(y)$]
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Simplify: ⓐ #math.equation(block: false, alt: "open parenthesis the square root of 6 x to the power 5 close parenthesis open parenthesis the square root of 2 x close parenthesis")[$( sqrt(6 x^(5)) ) ( sqrt(2 x) )$] ⓑ #math.equation(block: false, alt: "open parenthesis the square root of 12 y squared close parenthesis open parenthesis the square root of 3 y to the power 5 close parenthesis")[$( sqrt(12 y^(2)) ) ( sqrt(3 y^(5)) )$].

#solutionbox[
ⓐ #math.equation(block: false, alt: "2 x cubed the square root of 3")[$2 x^(3) sqrt(3)$] ⓑ #math.equation(block: false, alt: "6 y cubed the square root of y")[$6 y^(3) sqrt(y)$]
]
]

#examplebox("Example 4")[][
Simplify: #math.equation(block: false, alt: "open parenthesis 10 the square root of 6 p cubed close parenthesis open parenthesis 3 the square root of 18 p close parenthesis")[$( 10 sqrt(6 p^(3)) ) ( 3 sqrt(18 p) )$].

#solutionbox[
#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([], [#math.equation(block: false, alt: "open parenthesis 10 the square root of 6 p cubed close parenthesis open parenthesis 3 the square root of 18 p close parenthesis")[$( 10 sqrt(6 p^(3)) ) ( 3 sqrt(18 p) )$]]),
  [Multiply.], [#math.equation(block: false, alt: "30 the square root of 108 p to the power 4")[$30 sqrt(108 p^(4))$]],
  [Simplify the radical.], [#math.equation(block: false, alt: "30 the square root of 36 p to the power 4 times the square root of 3")[$30 sqrt(36 p^(4)) · sqrt(3)$]],
  [], [#math.equation(block: false, alt: "30 times 6 p squared times the square root of 3")[$30 · 6 p^(2) · sqrt(3)$]],
  [], [#math.equation(block: false, alt: "180 p squared the square root of 3")[$180 p^(2) sqrt(3)$]],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Simplify: #math.equation(block: false, alt: "open parenthesis 6 the square root of 2 x squared close parenthesis open parenthesis 8 the square root of 45 x to the power 4 close parenthesis")[$( 6 sqrt(2 x^(2)) ) ( 8 sqrt(45 x^(4)) )$].

#solutionbox[
#math.equation(block: true, alt: "144 x cubed the square root of 10")[$144 x^(3) sqrt(10)$]
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Simplify: #math.equation(block: false, alt: "open parenthesis 2 the square root of 6 y to the power 4 close parenthesis open parenthesis 12 the square root of 30 y close parenthesis")[$( 2 sqrt(6 y^(4)) ) ( 12 sqrt(30 y) )$].

#solutionbox[
#math.equation(block: true, alt: "144 y squared the square root of 5 y")[$144 y^(2) sqrt(5 y)$]
]
]

#examplebox("Example 5")[][
Simplify: ⓐ #math.equation(block: false, alt: "open parenthesis the square root of 2 close parenthesis squared")[$attach(( sqrt(2) ), t: 2)$] ⓑ #math.equation(block: false, alt: "open parenthesis − the square root of 11 close parenthesis squared")[$attach(( "−" sqrt(11) ), t: 2)$].

#solutionbox[
ⓐ  #linebreak()

#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([], [#math.equation(block: false, alt: "open parenthesis the square root of 2 close parenthesis squared")[$attach(( sqrt(2) ), t: 2)$]]),
  [Rewrite as a product.], [#math.equation(block: false, alt: "open parenthesis the square root of 2 close parenthesis open parenthesis the square root of 2 close parenthesis")[$( sqrt(2) ) ( sqrt(2) )$]],
  [Multiply.], [#math.equation(block: false, alt: "the square root of 4")[$sqrt(4)$]],
  [Simplify.], [#math.equation(block: false, alt: "2")[$2$]],
))

ⓑ  #linebreak()

#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([], [#math.equation(block: false, alt: "open parenthesis − the square root of 11 close parenthesis squared")[$attach(( "−" sqrt(11) ), t: 2)$]]),
  [Rewrite as a product.], [#math.equation(block: false, alt: "open parenthesis − the square root of 11 close parenthesis open parenthesis − the square root of 11 close parenthesis")[$( "−" sqrt(11) ) ( "−" sqrt(11) )$]],
  [Multiply.], [#math.equation(block: false, alt: "the square root of 121")[$sqrt(121)$]],
  [Simplify.], [#math.equation(block: false, alt: "11")[$11$]],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Simplify: ⓐ #math.equation(block: false, alt: "open parenthesis the square root of 12 close parenthesis squared")[$attach(( sqrt(12) ), t: 2)$] ⓑ #math.equation(block: false, alt: "open parenthesis − the square root of 15 close parenthesis squared")[$attach(( "−" sqrt(15) ), t: 2)$].

#solutionbox[
ⓐ 12 ⓑ #math.equation(block: false, alt: "15")[$15$]
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Simplify: ⓐ #math.equation(block: false, alt: "open parenthesis the square root of 16 close parenthesis squared")[$attach(( sqrt(16) ), t: 2)$] ⓑ #math.equation(block: false, alt: "open parenthesis − the square root of 20 close parenthesis squared")[$attach(( "−" sqrt(20) ), t: 2)$].

#solutionbox[
ⓐ 16 ⓑ 20
]
]

The results of the previous example lead us to this property.

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
#emph[Squaring a Square Root]

If #emph[a] is a nonnegative real number, then

#math.equation(block: true, alt: "open parenthesis the square root of a close parenthesis squared equals a")[$attach(( sqrt(a) ), t: 2) = a$]
]

By realizing that squaring and taking a square root are ‘opposite’ operations, we can simplify #math.equation(block: false, alt: "open parenthesis the square root of 2 close parenthesis squared")[$attach(( sqrt(2) ), t: 2)$] and get 2 right away. When we multiply the two #strong[like square roots] in part (a) of the next example, it is the same as squaring.

#examplebox("Example 6")[][
Simplify: ⓐ #math.equation(block: false, alt: "open parenthesis 2 the square root of 3 close parenthesis open parenthesis 8 the square root of 3 close parenthesis")[$( 2 sqrt(3) ) ( 8 sqrt(3) )$] ⓑ #math.equation(block: false, alt: "open parenthesis 3 the square root of 6 close parenthesis squared")[$attach(( 3 sqrt(6) ), t: 2)$].

#solutionbox[
ⓐ  #linebreak()

#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([], [#math.equation(block: false, alt: "open parenthesis 2 the square root of 3 close parenthesis open parenthesis 8 the square root of 3 close parenthesis")[$( 2 sqrt(3) ) ( 8 sqrt(3) )$]]),
  [Multiply. Remember, #math.equation(block: false, alt: "open parenthesis the square root of 3 close parenthesis squared equals 3 .")[$attach(( sqrt(3) ), t: 2) = 3 .$]], [#math.equation(block: false, alt: "16 times 3")[$16 · 3$]],
  [Simplify.], [#math.equation(block: false, alt: "48")[$48$]],
))

ⓑ  #linebreak()

#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([], [#math.equation(block: false, alt: "open parenthesis 3 the square root of 6 close parenthesis squared")[$attach(( 3 sqrt(6) ), t: 2)$]]),
  [Multiply.], [#math.equation(block: false, alt: "9 times 6")[$9 · 6$]],
  [Simplify.], [#math.equation(block: false, alt: "54")[$54$]],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Simplify: ⓐ #math.equation(block: false, alt: "open parenthesis 6 the square root of 11 close parenthesis open parenthesis 5 the square root of 11 close parenthesis")[$( 6 sqrt(11) ) ( 5 sqrt(11) )$] ⓑ #math.equation(block: false, alt: "open parenthesis 5 the square root of 8 close parenthesis squared")[$attach(( 5 sqrt(8) ), t: 2)$].

#solutionbox[
ⓐ 330 ⓑ 200
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Simplify: ⓐ #math.equation(block: false, alt: "open parenthesis 3 the square root of 7 close parenthesis open parenthesis 10 the square root of 7 close parenthesis")[$( 3 sqrt(7) ) ( 10 sqrt(7) )$] ⓑ #math.equation(block: false, alt: "open parenthesis −4 the square root of 6 close parenthesis squared")[$attach(( −4 sqrt(6) ), t: 2)$].

#solutionbox[
ⓐ 210 ⓑ 96
]
]


=== Use Polynomial Multiplication to Multiply Square Roots

In the next few examples, we will use the Distributive Property to multiply expressions with square roots.

We will first distribute and then simplify the square roots when possible.

#examplebox("Example 7")[][
Simplify: ⓐ #math.equation(block: false, alt: "3 open parenthesis 5 minus the square root of 2 close parenthesis")[$3 ( 5 − sqrt(2) )$] ⓑ #math.equation(block: false, alt: "the square root of 2 open parenthesis 4 minus the square root of 10 close parenthesis")[$sqrt(2) ( 4 − sqrt(10) )$].

#solutionbox[
ⓐ  #linebreak()

#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([], [#math.equation(block: false, alt: "3 open parenthesis 5 minus the square root of 2 close parenthesis")[$3 ( 5 − sqrt(2) )$]]),
  [Distribute.], [#math.equation(block: false, alt: "15 minus 3 the square root of 2")[$15 − 3 sqrt(2)$]],
))

ⓑ  #linebreak()

#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([], [#math.equation(block: false, alt: "the square root of 2 open parenthesis 4 minus the square root of 10 close parenthesis")[$sqrt(2) ( 4 − sqrt(10) )$]]),
  [Distribute.], [#math.equation(block: false, alt: "4 the square root of 2 minus the square root of 20")[$4 sqrt(2) − sqrt(20)$]],
  [], [#math.equation(block: false, alt: "4 the square root of 2 minus 2 the square root of 5")[$4 sqrt(2) − 2 sqrt(5)$]],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Simplify: ⓐ #math.equation(block: false, alt: "2 open parenthesis 3 minus the square root of 5 close parenthesis")[$2 ( 3 − sqrt(5) )$] ⓑ #math.equation(block: false, alt: "the square root of 3 open parenthesis 2 minus the square root of 18 close parenthesis")[$sqrt(3) ( 2 − sqrt(18) )$].

#solutionbox[
ⓐ #math.equation(block: false, alt: "6 minus 2 the square root of 5")[$6 − 2 sqrt(5)$] ⓑ #math.equation(block: false, alt: "2 the square root of 3 minus 3 the square root of 6")[$2 sqrt(3) − 3 sqrt(6)$]
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Simplify: ⓐ #math.equation(block: false, alt: "6 open parenthesis 2 plus the square root of 6 close parenthesis")[$6 ( 2 + sqrt(6) )$] ⓑ #math.equation(block: false, alt: "the square root of 7 open parenthesis 1 plus the square root of 14 close parenthesis")[$sqrt(7) ( 1 + sqrt(14) )$].

#solutionbox[
ⓐ #math.equation(block: false, alt: "12 plus 6 the square root of 6")[$12 + 6 sqrt(6)$] ⓑ #math.equation(block: false, alt: "the square root of 7 plus 7 the square root of 2")[$sqrt(7) + 7 sqrt(2)$]
]
]

#examplebox("Example 8")[][
Simplify: ⓐ #math.equation(block: false, alt: "the square root of 5 open parenthesis 7 plus 2 the square root of 5 close parenthesis")[$sqrt(5) ( 7 + 2 sqrt(5) )$] ⓑ #math.equation(block: false, alt: "the square root of 6 open parenthesis the square root of 2 plus the square root of 18 close parenthesis")[$sqrt(6) ( sqrt(2) + sqrt(18) )$].

#solutionbox[
ⓐ  #linebreak()

#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([], [#math.equation(block: false, alt: "the square root of 5 open parenthesis 7 plus 2 the square root of 5 close parenthesis")[$sqrt(5) ( 7 + 2 sqrt(5) )$]]),
  [Multiply.], [#math.equation(block: false, alt: "7 the square root of 5 plus 2 times 5")[$7 sqrt(5) + 2 · 5$]],
  [Simplify.], [#math.equation(block: false, alt: "7 the square root of 5 plus 10")[$7 sqrt(5) + 10$]],
  [], [#math.equation(block: false, alt: "10 plus 7 the square root of 5")[$10 + 7 sqrt(5)$]],
))

ⓑ  #linebreak()

#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([], [#math.equation(block: false, alt: "the square root of 6 open parenthesis the square root of 2 plus the square root of 18 close parenthesis")[$sqrt(6) ( sqrt(2) + sqrt(18) )$]]),
  [Multiply.], [#math.equation(block: false, alt: "the square root of 12 plus the square root of 108")[$sqrt(12) + sqrt(108)$]],
  [Simplify.], [#math.equation(block: false, alt: "the square root of 4 times the square root of 3 plus the square root of 36 times the square root of 3")[$sqrt(4) · sqrt(3) + sqrt(36) · sqrt(3)$]],
  [], [#math.equation(block: false, alt: "2 the square root of 3 plus 6 the square root of 3")[$2 sqrt(3) + 6 sqrt(3)$]],
  [Combine like radicals.], [#math.equation(block: false, alt: "8 the square root of 3")[$8 sqrt(3)$]],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Simplify: ⓐ #math.equation(block: false, alt: "the square root of 6 open parenthesis 1 plus 3 the square root of 6 close parenthesis")[$sqrt(6) ( 1 + 3 sqrt(6) )$] ⓑ #math.equation(block: false, alt: "the square root of 12 open parenthesis the square root of 3 plus the square root of 24 close parenthesis")[$sqrt(12) ( sqrt(3) + sqrt(24) )$].

#solutionbox[
ⓐ #math.equation(block: false, alt: "18 plus the square root of 6")[$18 + sqrt(6)$] ⓑ #math.equation(block: false, alt: "6 plus 12 the square root of 2")[$6 + 12 sqrt(2)$]
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Simplify: ⓐ #math.equation(block: false, alt: "the square root of 8 open parenthesis 2 minus 5 the square root of 8 close parenthesis")[$sqrt(8) ( 2 − 5 sqrt(8) )$] ⓑ #math.equation(block: false, alt: "the square root of 14 open parenthesis the square root of 2 plus the square root of 42 close parenthesis")[$sqrt(14) ( sqrt(2) + sqrt(42) )$].

#solutionbox[
ⓐ #math.equation(block: false, alt: "−40 plus 4 the square root of 2")[$−40 + 4 sqrt(2)$] ⓑ #math.equation(block: false, alt: "2 the square root of 7 plus 14 the square root of 3")[$2 sqrt(7) + 14 sqrt(3)$]
]
]

When we worked with polynomials, we multiplied binomials by binomials. Remember, this gave us four products before we combined any like terms. To be sure to get all four products, we organized our work—usually by the FOIL method.

#examplebox("Example 9")[][
Simplify: #math.equation(block: false, alt: "open parenthesis 2 plus the square root of 3 close parenthesis open parenthesis 4 minus the square root of 3 close parenthesis")[$( 2 + sqrt(3) ) ( 4 − sqrt(3) )$].

#solutionbox[
#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([], [#math.equation(block: false, alt: "open parenthesis 2 plus the square root of 3 close parenthesis open parenthesis 4 minus the square root of 3 close parenthesis")[$( 2 + sqrt(3) ) ( 4 − sqrt(3) )$]]),
  [Multiply.], [#math.equation(block: false, alt: "8 minus 2 the square root of 3 plus 4 the square root of 3 minus 3")[$8 − 2 sqrt(3) + 4 sqrt(3) − 3$]],
  [Combine like terms.], [#math.equation(block: false, alt: "5 plus 2 the square root of 3")[$5 + 2 sqrt(3)$]],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Simplify: #math.equation(block: false, alt: "open parenthesis 1 plus the square root of 6 close parenthesis open parenthesis 3 minus the square root of 6 close parenthesis")[$( 1 + sqrt(6) ) ( 3 − sqrt(6) )$].

#solutionbox[
#math.equation(block: true, alt: "−3 plus 2 the square root of 6")[$−3 + 2 sqrt(6)$]
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Simplify: #math.equation(block: false, alt: "open parenthesis 4 minus the square root of 10 close parenthesis open parenthesis 2 plus the square root of 10 close parenthesis")[$( 4 − sqrt(10) ) ( 2 + sqrt(10) )$].

#solutionbox[
#math.equation(block: true, alt: "−2 plus 2 the square root of 10")[$−2 + 2 sqrt(10)$]
]
]

#examplebox("Example 10")[][
Simplify: #math.equation(block: false, alt: "open parenthesis 3 minus 2 the square root of 7 close parenthesis open parenthesis 4 minus 2 the square root of 7 close parenthesis")[$( 3 − 2 sqrt(7) ) ( 4 − 2 sqrt(7) )$].

#solutionbox[
#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([], [#math.equation(block: false, alt: "open parenthesis 3 minus 2 the square root of 7 close parenthesis open parenthesis 4 minus 2 the square root of 7 close parenthesis")[$( 3 − 2 sqrt(7) ) ( 4 − 2 sqrt(7) )$]]),
  [Multiply.], [#math.equation(block: false, alt: "12 minus 6 the square root of 7 minus 8 the square root of 7 plus 4 times 7")[$12 − 6 sqrt(7) − 8 sqrt(7) + 4 · 7$]],
  [Simplify.], [#math.equation(block: false, alt: "12 minus 6 the square root of 7 minus 8 the square root of 7 plus 28")[$12 − 6 sqrt(7) − 8 sqrt(7) + 28$]],
  [Combine like terms.], [#math.equation(block: false, alt: "40 minus 14 the square root of 7")[$40 − 14 sqrt(7)$]],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Simplify: #math.equation(block: false, alt: "open parenthesis 6 minus 3 the square root of 7 close parenthesis open parenthesis 3 plus 4 the square root of 7 close parenthesis")[$( 6 − 3 sqrt(7) ) ( 3 + 4 sqrt(7) )$].

#solutionbox[
#math.equation(block: true, alt: "−66 plus 15 the square root of 7")[$−66 + 15 sqrt(7)$]
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Simplify: #math.equation(block: false, alt: "open parenthesis 2 minus 3 the square root of 11 close parenthesis open parenthesis 4 minus the square root of 11 close parenthesis")[$( 2 − 3 sqrt(11) ) ( 4 − sqrt(11) )$].

#solutionbox[
#math.equation(block: true, alt: "41 minus 14 the square root of 11")[$41 − 14 sqrt(11)$]
]
]

#examplebox("Example 11")[][
Simplify: #math.equation(block: false, alt: "open parenthesis 3 the square root of 2 minus the square root of 5 close parenthesis open parenthesis the square root of 2 plus 4 the square root of 5 close parenthesis")[$( 3 sqrt(2) − sqrt(5) ) ( sqrt(2) + 4 sqrt(5) )$].

#solutionbox[
#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([], [#math.equation(block: false, alt: "open parenthesis 3 the square root of 2 minus the square root of 5 close parenthesis open parenthesis the square root of 2 plus 4 the square root of 5 close parenthesis")[$( 3 sqrt(2) − sqrt(5) ) ( sqrt(2) + 4 sqrt(5) )$]]),
  [Multiply.], [#math.equation(block: false, alt: "3 times 2 plus 12 the square root of 10 minus the square root of 10 minus 4 times 5")[$3 · 2 + 12 sqrt(10) − sqrt(10) − 4 · 5$]],
  [Simplify.], [#math.equation(block: false, alt: "6 plus 12 the square root of 10 minus the square root of 10 minus 20")[$6 + 12 sqrt(10) − sqrt(10) − 20$]],
  [Combine like terms.], [#math.equation(block: false, alt: "−14 plus 11 the square root of 10")[$−14 + 11 sqrt(10)$]],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Simplify: #math.equation(block: false, alt: "open parenthesis 5 the square root of 3 minus the square root of 7 close parenthesis open parenthesis the square root of 3 plus 2 the square root of 7 close parenthesis")[$( 5 sqrt(3) − sqrt(7) ) ( sqrt(3) + 2 sqrt(7) )$].

#solutionbox[
#math.equation(block: true, alt: "1 plus 9 the square root of 21")[$1 + 9 sqrt(21)$]
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Simplify: #math.equation(block: false, alt: "open parenthesis the square root of 6 minus 3 the square root of 8 close parenthesis open parenthesis 2 the square root of 6 plus the square root of 8 close parenthesis")[$( sqrt(6) − 3 sqrt(8) ) ( 2 sqrt(6) + sqrt(8) )$]

#solutionbox[
#math.equation(block: true, alt: "−12 minus 20 the square root of 3")[$−12 − 20 sqrt(3)$]
]
]

#examplebox("Example 12")[][
Simplify: #math.equation(block: false, alt: "open parenthesis 4 minus 2 the square root of x close parenthesis open parenthesis 1 plus 3 the square root of x close parenthesis")[$( 4 − 2 sqrt(x) ) ( 1 + 3 sqrt(x) )$].

#solutionbox[
#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([], [#math.equation(block: false, alt: "open parenthesis 4 minus 2 the square root of x close parenthesis open parenthesis 1 plus 3 the square root of x close parenthesis")[$( 4 − 2 sqrt(x) ) ( 1 + 3 sqrt(x) )$]]),
  [Multiply.], [#math.equation(block: false, alt: "4 plus 12 the square root of x minus 2 the square root of x minus 6 x")[$4 + 12 sqrt(x) − 2 sqrt(x) − 6 x$]],
  [Combine like terms.], [#math.equation(block: false, alt: "4 plus 10 the square root of x minus 6 x")[$4 + 10 sqrt(x) − 6 x$]],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Simplify: #math.equation(block: false, alt: "open parenthesis 6 minus 5 the square root of m close parenthesis open parenthesis 2 plus 3 the square root of m close parenthesis")[$( 6 − 5 sqrt(m) ) ( 2 + 3 sqrt(m) )$].

#solutionbox[
#math.equation(block: true, alt: "12 plus 8 the square root of m minus 15 m")[$12 + 8 sqrt(m) − 15 m$]
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Simplify: #math.equation(block: false, alt: "open parenthesis 10 plus 3 the square root of n close parenthesis open parenthesis 1 minus 5 the square root of n close parenthesis")[$( 10 + 3 sqrt(n) ) ( 1 − 5 sqrt(n) )$].

#solutionbox[
#math.equation(block: true, alt: "10 minus 47 the square root of n minus 15 n")[$10 − 47 sqrt(n) − 15 n$]
]
]

Note that some special products made our work easier when we multiplied binomials earlier. This is true when we multiply square roots, too. The special product formulas we used are shown below.

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
#emph[Special Product Formulas]

#math.equation(block: true, alt: "Binomial Squares, Product of Conjugates; open parenthesis a plus b close parenthesis squared equals a squared plus 2 a b plus b squared, open parenthesis a minus b close parenthesis open parenthesis a plus b close parenthesis equals a squared minus b squared; open parenthesis a minus b close parenthesis squared equals a squared minus 2 a b plus b squared")[$"Binomial Squares" &  &  & #h(4em) "Product of Conjugates" \ attach(( a + b ), t: 2) = a^(2) + 2 a b + b^(2) &  &  & #h(4em) ( a − b ) ( a + b ) = a^(2) − b^(2) \  \  \ attach(( a − b ), t: 2) = a^(2) − 2 a b + b^(2) &  &  &$]
]

We will use the special product formulas in the next few examples. We will start with the Binomial Squares formula.

#examplebox("Example 13")[][
Simplify: ⓐ #math.equation(block: false, alt: "open parenthesis 2 plus the square root of 3 close parenthesis squared")[$attach(( 2 + sqrt(3) ), t: 2)$] ⓑ #math.equation(block: false, alt: "open parenthesis 4 minus 2 the square root of 5 close parenthesis squared")[$attach(( 4 − 2 sqrt(5) ), t: 2)$].

#solutionbox[
Be sure to include the #math.equation(block: false, alt: "2 a b")[$2 a b$] term when squaring a binomial.

+ ⓐ  #linebreak() 
#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([], [#figure(figph[Algebraic identity (a + b)² in red, contrasted with its numerical application (2 + √3)² in black, showcasing the square of a sum.], alt: "Algebraic identity (a + b)² in red, contrasted with its numerical application (2 + √3)² in black, showcasing the square of a sum.", caption: none)]),
  [Multiply using the binomial square pattern.], [#figure(figph[A mathematical expression illustrating the algebraic identity (a+b)^2 = a^2 + 2ab + b^2, with 'a' represented by 2 and 'b' by the square root of 3.], alt: "A mathematical expression illustrating the algebraic identity (a+b)^2 = a^2 + 2ab + b^2, with 'a' represented by 2 and 'b' by the square root of 3.", caption: none)],
  [Simplify.], [#figure(figph[The image displays the mathematical expression: 4 + 4√3 + 3, which includes integers and a radical term.], alt: "The image displays the mathematical expression: 4 + 4√3 + 3, which includes integers and a radical term.", caption: none)],
  [Combine like terms.], [#figure(figph[The image displays the mathematical expression '7 + 4 times the square root of 3' on a white background. The numbers and symbols are clear and centrally located.], alt: "The image displays the mathematical expression '7 + 4 times the square root of 3' on a white background. The numbers and symbols are clear and centrally located.", caption: none)],
))
+ ⓑ  #linebreak() 
#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([], [#figure(figph[Mathematical expressions for (a-b)^2 in red and (4-2√5)^2 in black, illustrating the application of an algebraic identity to a numerical problem.], alt: "Mathematical expressions for (a-b)^2 in red and (4-2√5)^2 in black, illustrating the application of an algebraic identity to a numerical problem.", caption: none)]),
  [Multiply using the binomial square pattern.], [#figure(figph[The algebraic identity a^2 - 2ab + b^2 illustrated with a=4 and b=2sqrt(5) in an expanded form.], alt: "The algebraic identity a^2 - 2ab + b^2 illustrated with a=4 and b=2sqrt(5) in an expanded form.", caption: none)],
  [Simplify.], [#figure(figph[A mathematical expression showing the simplification of 16 - 16√5 + 4 \* 5 to 16 - 16√5 + 20, where the multiplication 4 \* 5 has been performed.], alt: "A mathematical expression showing the simplification of 16 - 16√5 + 4 * 5 to 16 - 16√5 + 20, where the multiplication 4 * 5 has been performed.", caption: none)],
  [Combine like terms.], [#figure(figph[A mathematical expression displaying 36 - 16 times the square root of 5 on a white background.], alt: "A mathematical expression displaying 36 - 16 times the square root of 5 on a white background.", caption: none)],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Simplify: ⓐ #math.equation(block: false, alt: "open parenthesis 10 plus the square root of 2 close parenthesis squared")[$attach(( 10 + sqrt(2) ), t: 2)$] ⓑ #math.equation(block: false, alt: "open parenthesis 1 plus 3 the square root of 6 close parenthesis squared")[$attach(( 1 + 3 sqrt(6) ), t: 2)$].

#solutionbox[
ⓐ #math.equation(block: false, alt: "102 plus 20 the square root of 2")[$102 + 20 sqrt(2)$] ⓑ #math.equation(block: false, alt: "55 plus 6 the square root of 6")[$55 + 6 sqrt(6)$]
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Simplify: ⓐ #math.equation(block: false, alt: "open parenthesis 6 minus the square root of 5 close parenthesis squared")[$attach(( 6 − sqrt(5) ), t: 2)$] ⓑ #math.equation(block: false, alt: "open parenthesis 9 minus 2 the square root of 10 close parenthesis squared")[$attach(( 9 − 2 sqrt(10) ), t: 2)$].

#solutionbox[
ⓐ #math.equation(block: false, alt: "41 minus 12 the square root of 5")[$41 − 12 sqrt(5)$] ⓑ #math.equation(block: false, alt: "121 minus 36 the square root of 10")[$121 − 36 sqrt(10)$]
]
]

#examplebox("Example 14")[][
Simplify: #math.equation(block: false, alt: "open parenthesis 1 plus 3 the square root of x close parenthesis squared")[$attach(( 1 + 3 sqrt(x) ), t: 2)$].

#solutionbox[
#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([], [#figure(figph[An image showcasing the algebraic identity (a+b)^2 in red, followed by a specific example (1+3√x)^2 in black, illustrating the square of a binomial.], alt: "An image showcasing the algebraic identity (a+b)^2 in red, followed by a specific example (1+3√x)^2 in black, illustrating the square of a binomial.", caption: none)]),
  [Multiply using the binomial square pattern.], [#figure(figph[An algebraic expression demonstrating the binomial square formula (a+b)^2, where a=1 and b=3sqrt(x) are substituted into the expansion a^2 + 2ab + b^2.], alt: "An algebraic expression demonstrating the binomial square formula (a+b)^2, where a=1 and b=3sqrt(x) are substituted into the expansion a^2 + 2ab + b^2.", caption: none)],
  [Simplify.], [#figure(figph[The mathematical expression '1 + 6√x + 9x' is displayed, an algebraic sum involving a constant, a term with the square root of x, and a term with x.], alt: "The mathematical expression '1 + 6√x + 9x' is displayed, an algebraic sum involving a constant, a term with the square root of x, and a term with x.", caption: none)],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Simplify: #math.equation(block: false, alt: "open parenthesis 2 plus 5 the square root of m close parenthesis squared")[$attach(( 2 + 5 sqrt(m) ), t: 2)$].

#solutionbox[
#math.equation(block: true, alt: "4 plus 20 the square root of m plus 25 m")[$4 + 20 sqrt(m) + 25 m$]
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Simplify: #math.equation(block: false, alt: "open parenthesis 3 minus 4 the square root of n close parenthesis squared")[$attach(( 3 − 4 sqrt(n) ), t: 2)$].

#solutionbox[
#math.equation(block: true, alt: "9 minus 24 the square root of n plus 16 n")[$9 − 24 sqrt(n) + 16 n$]
]
]

In the next two examples, we will find the product of conjugates.

#examplebox("Example 15")[][
Simplify: #math.equation(block: false, alt: "open parenthesis 4 minus the square root of 2 close parenthesis open parenthesis 4 plus the square root of 2 close parenthesis")[$( 4 − sqrt(2) ) ( 4 + sqrt(2) )$].

#solutionbox[
#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([], [#figure(figph[The difference of squares formula, a minus b times a plus b equals a squared minus b squared, is shown. The example uses 4 minus the square root of 2 and 4 plus the square root of 2.], alt: "The difference of squares formula, a minus b times a plus b equals a squared minus b squared, is shown. The example uses 4 minus the square root of 2 and 4 plus the square root of 2.", caption: none)]),
  [Multiply using the binomial square pattern.], [#figure(figph[The difference of squares formula, a^2 - b^2, applied to 4^2 - (sqrt(2))^2.], alt: "The difference of squares formula, a^2 - b^2, applied to 4^2 - (sqrt(2))^2.", caption: none)],
  [Simplify.], [#figure(figph[A simple subtraction problem with the numbers 16 minus 2, equaling 14, displayed vertically on a white background.], alt: "A simple subtraction problem with the numbers 16 minus 2, equaling 14, displayed vertically on a white background.", caption: none)],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Simplify: #math.equation(block: false, alt: "open parenthesis 2 minus the square root of 3 close parenthesis open parenthesis 2 plus the square root of 3 close parenthesis")[$( 2 − sqrt(3) ) ( 2 + sqrt(3) )$].

#solutionbox[
#math.equation(block: true, alt: "1")[$1$]
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Simplify: #math.equation(block: false, alt: "open parenthesis 1 plus the square root of 5 close parenthesis open parenthesis 1 minus the square root of 5 close parenthesis")[$( 1 + sqrt(5) ) ( 1 − sqrt(5) )$].

#solutionbox[
#math.equation(block: true, alt: "−4")[$−4$]
]
]

#examplebox("Example 16")[][
Simplify: #math.equation(block: false, alt: "open parenthesis 5 minus 2 the square root of 3 close parenthesis open parenthesis 5 plus 2 the square root of 3 close parenthesis")[$( 5 − 2 sqrt(3) ) ( 5 + 2 sqrt(3) )$].

#solutionbox[
#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([], [#figure(figph[This image highlights the difference of squares formula, (a - b)(a + b), demonstrating its application with the numerical example (5 - 2√3)(5 + 2√3).], alt: "This image highlights the difference of squares formula, (a - b)(a + b), demonstrating its application with the numerical example (5 - 2√3)(5 + 2√3).", caption: none)]),
  [Multiply using the binomial square pattern.], [#figure(figph[A mathematical expression displaying the difference of two squares. The formula a^2 - b^2 is shown above the specific calculation 5^2 - (2√3)^2, illustrating its application.], alt: "A mathematical expression displaying the difference of two squares. The formula a^2 - b^2 is shown above the specific calculation 5^2 - (2√3)^2, illustrating its application.", caption: none)],
  [Simplify.], [#figure(figph[A fraction displaying the operation (25 - 4 \* 3) divided by 13. Following the order of operations, the numerator simplifies to 25 - 12 = 13, making the fraction 13/13, which equals 1.], alt: "A fraction displaying the operation (25 - 4 * 3) divided by 13. Following the order of operations, the numerator simplifies to 25 - 12 = 13, making the fraction 13/13, which equals 1.", caption: none)],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Simplify: #math.equation(block: false, alt: "open parenthesis 3 minus 2 the square root of 5 close parenthesis open parenthesis 3 plus 2 the square root of 5 close parenthesis")[$( 3 − 2 sqrt(5) ) ( 3 + 2 sqrt(5) )$].

#solutionbox[
#math.equation(block: true, alt: "−11")[$−11$]
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Simplify: #math.equation(block: false, alt: "open parenthesis 4 plus 5 the square root of 7 close parenthesis open parenthesis 4 minus 5 the square root of 7 close parenthesis")[$( 4 + 5 sqrt(7) ) ( 4 − 5 sqrt(7) )$].

#solutionbox[
#math.equation(block: true, alt: "−159")[$−159$]
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Access these online resources for additional instruction and practice with multiplying square roots.

- #link("https://openstax.org/l/25ProductProp")[Product Property]
- #link("https://openstax.org/l/25MultPolySR")[Multiply Binomials with Square Roots]
]


=== Key Concepts

- #strong[Product Property of Square Roots] If #emph[a], #emph[b] are nonnegative real numbers, then #linebreak() 
#math.equation(block: true, alt: "the square root of a b equals the square root of a times the square root of b and the square root of a times the square root of b equals the square root of a b")[$sqrt(a b) = sqrt(a) · sqrt(b) #h(1em) "and" #h(1em) sqrt(a) · sqrt(b) = sqrt(a b)$]
- #strong[Special formulas] for multiplying binomials and conjugates: #linebreak() 
#math.equation(block: true, alt: "open parenthesis a plus b close parenthesis squared equals a squared plus 2 a b plus b squared, open parenthesis a minus b close parenthesis open parenthesis a plus b close parenthesis equals a squared minus b squared; open parenthesis a minus b close parenthesis squared equals a squared minus 2 a b plus b squared")[$attach(( a + b ), t: 2) = a^(2) + 2 a b + b^(2) &  &  & ( a − b ) ( a + b ) = a^(2) − b^(2) \ attach(( a − b ), t: 2) = a^(2) − 2 a b + b^(2) &  &  &$]
- The FOIL method can be used to multiply binomials containing radicals.


==== Practice Makes Perfect

#strong[Multiply Square Roots]

In the following exercises, simplify.

ⓐ #math.equation(block: false, alt: "the square root of 2 times the square root of 8")[$sqrt(2) · sqrt(8)$] ⓑ #math.equation(block: false, alt: "open parenthesis 3 the square root of 3 close parenthesis open parenthesis 2 the square root of 18 close parenthesis")[$( 3 sqrt(3) ) ( 2 sqrt(18) )$]

#solutionbox[
ⓐ #math.equation(block: false, alt: "4")[$4$] ⓑ #math.equation(block: false, alt: "18 the square root of 6")[$18 sqrt(6)$]
]

ⓐ #math.equation(block: false, alt: "the square root of 6 times the square root of 6")[$sqrt(6) · sqrt(6)$] ⓑ #math.equation(block: false, alt: "open parenthesis 3 the square root of 2 close parenthesis open parenthesis 2 the square root of 32 close parenthesis")[$( 3 sqrt(2) ) ( 2 sqrt(32) )$]

ⓐ #math.equation(block: false, alt: "the square root of 7 times the square root of 14")[$sqrt(7) · sqrt(14)$] ⓑ #math.equation(block: false, alt: "open parenthesis 4 the square root of 8 close parenthesis open parenthesis 5 the square root of 8 close parenthesis")[$( 4 sqrt(8) ) ( 5 sqrt(8) )$]

#solutionbox[
ⓐ #math.equation(block: false, alt: "7 the square root of 2")[$7 sqrt(2)$] ⓑ 160
]

ⓐ #math.equation(block: false, alt: "the square root of 6 times the square root of 12")[$sqrt(6) · sqrt(12)$] ⓑ #math.equation(block: false, alt: "open parenthesis 2 the square root of 5 close parenthesis open parenthesis 2 the square root of 10 close parenthesis")[$( 2 sqrt(5) ) ( 2 sqrt(10) )$]

#math.equation(block: true, alt: "open parenthesis 5 the square root of 2 close parenthesis open parenthesis 3 the square root of 6 close parenthesis")[$( 5 sqrt(2) ) ( 3 sqrt(6) )$]

#solutionbox[
#math.equation(block: true, alt: "30 the square root of 3")[$30 sqrt(3)$]
]

#math.equation(block: true, alt: "open parenthesis 2 the square root of 3 close parenthesis open parenthesis 4 the square root of 6 close parenthesis")[$( 2 sqrt(3) ) ( 4 sqrt(6) )$]

#math.equation(block: true, alt: "open parenthesis −2 the square root of 3 close parenthesis open parenthesis 3 the square root of 18 close parenthesis")[$( −2 sqrt(3) ) ( 3 sqrt(18) )$]

#solutionbox[
#math.equation(block: true, alt: "−18 the square root of 6")[$−18 sqrt(6)$]
]

#math.equation(block: true, alt: "open parenthesis −4 the square root of 5 close parenthesis open parenthesis 5 the square root of 10 close parenthesis")[$( −4 sqrt(5) ) ( 5 sqrt(10) )$]

#math.equation(block: true, alt: "open parenthesis 5 the square root of 6 close parenthesis open parenthesis − the square root of 12 close parenthesis")[$( 5 sqrt(6) ) ( "−" sqrt(12) )$]

#solutionbox[
#math.equation(block: true, alt: "−30 the square root of 2")[$−30 sqrt(2)$]
]

#math.equation(block: true, alt: "open parenthesis 6 the square root of 2 close parenthesis open parenthesis − the square root of 10 close parenthesis")[$( 6 sqrt(2) ) ( "−" sqrt(10) )$]

#math.equation(block: true, alt: "open parenthesis −2 the square root of 7 close parenthesis open parenthesis −2 the square root of 14 close parenthesis")[$( −2 sqrt(7) ) ( −2 sqrt(14) )$]

#solutionbox[
#math.equation(block: true, alt: "28 the square root of 2")[$28 sqrt(2)$]
]

#math.equation(block: true, alt: "open parenthesis −2 the square root of 11 close parenthesis open parenthesis −4 the square root of 22 close parenthesis")[$( −2 sqrt(11) ) ( −4 sqrt(22) )$]

ⓐ #math.equation(block: false, alt: "open parenthesis the square root of 15 y close parenthesis open parenthesis the square root of 5 y cubed close parenthesis")[$( sqrt(15 y) ) ( sqrt(5 y^(3)) )$] ⓑ #math.equation(block: false, alt: "open parenthesis the square root of 2 n squared close parenthesis open parenthesis the square root of 18 n cubed close parenthesis")[$( sqrt(2 n^(2)) ) ( sqrt(18 n^(3)) )$]

#solutionbox[
ⓐ #math.equation(block: false, alt: "5 y squared the square root of 3")[$5 y^(2) sqrt(3)$] ⓑ #math.equation(block: false, alt: "6 n squared the square root of n")[$6 n^(2) sqrt(n)$]
]

ⓐ #math.equation(block: false, alt: "open parenthesis the square root of 14 x cubed close parenthesis open parenthesis the square root of 7 x cubed close parenthesis")[$( sqrt(14 x^(3)) ) ( sqrt(7 x^(3)) )$] ⓑ #math.equation(block: false, alt: "open parenthesis the square root of 3 q squared close parenthesis open parenthesis the square root of 48 q cubed close parenthesis")[$( sqrt(3 q^(2)) ) ( sqrt(48 q^(3)) )$]

ⓐ #math.equation(block: false, alt: "open parenthesis the square root of 16 y squared close parenthesis open parenthesis the square root of 8 y to the power 4 close parenthesis")[$( sqrt(16 y^(2)) ) ( sqrt(8 y^(4)) )$] ⓑ #math.equation(block: false, alt: "open parenthesis the square root of 11 s to the power 6 close parenthesis open parenthesis the square root of 11 s close parenthesis")[$( sqrt(11 s^(6)) ) ( sqrt(11 s) )$]

#solutionbox[
ⓐ #math.equation(block: false, alt: "8 y cubed the square root of 2")[$8 y^(3) sqrt(2)$] ⓑ #math.equation(block: false, alt: "11 s cubed the square root of s")[$11 s^(3) sqrt(s)$]
]

ⓐ #math.equation(block: false, alt: "open parenthesis the square root of 8 x cubed close parenthesis open parenthesis the square root of 3 x close parenthesis")[$( sqrt(8 x^(3)) ) ( sqrt(3 x) )$] ⓑ #math.equation(block: false, alt: "open parenthesis the square root of 7 r close parenthesis open parenthesis the square root of 7 r to the power 8 close parenthesis")[$( sqrt(7 r) ) ( sqrt(7 r^(8)) )$]

#math.equation(block: true, alt: "open parenthesis 2 the square root of 5 b cubed close parenthesis open parenthesis 4 the square root of 15 b close parenthesis")[$( 2 sqrt(5 b^(3)) ) ( 4 sqrt(15 b) )$]

#solutionbox[
#math.equation(block: true, alt: "40 b squared the square root of 3")[$40 b^(2) sqrt(3)$]
]

#math.equation(block: true, alt: "open parenthesis 3 the square root of 8 c to the power 5 close parenthesis open parenthesis 2 the square root of 6 c cubed close parenthesis")[$( 3 sqrt(8 c^(5)) ) ( 2 sqrt(6 c^(3)) )$]

#math.equation(block: true, alt: "open parenthesis 5 the square root of 2 d to the power 7 close parenthesis open parenthesis 3 the square root of 50 d cubed close parenthesis")[$( 5 sqrt(2 d^(7)) ) ( 3 sqrt(50 d^(3)) )$]

#solutionbox[
#math.equation(block: true, alt: "150 d to the power 5")[$150 d^(5)$]
]

#math.equation(block: true, alt: "4 the square root of 6 t squared 3 the square root of 3 t squared")[$4 sqrt(6 t^(2)) 3 sqrt(3 t^(2))$]

#math.equation(block: true, alt: "3 the square root of 4 y to the power 4 3 the square root of 9 y to the power 5")[$3 sqrt(4 y^(4)) 3 sqrt(9 y^(5))$]

#solutionbox[
#math.equation(block: true, alt: "54 y to the power 4 the square root of y")[$54 y^(4) sqrt(y)$]
]

#math.equation(block: true, alt: "open parenthesis −2 the square root of 7 z cubed close parenthesis open parenthesis 3 the square root of 14 z to the power 8 close parenthesis")[$( −2 sqrt(7 z^(3)) ) ( 3 sqrt(14 z^(8)) )$]

#math.equation(block: true, alt: "open parenthesis 4 the square root of 2 k to the power 5 close parenthesis open parenthesis −3 the square root of 32 k to the power 6 close parenthesis")[$( 4 sqrt(2 k^(5)) ) ( −3 sqrt(32 k^(6)) )$]

#solutionbox[
#math.equation(block: true, alt: "−96 k to the power 5 the square root of k")[$−96 k^(5) sqrt(k)$]
]

ⓐ #math.equation(block: false, alt: "open parenthesis the square root of 7 close parenthesis squared")[$attach(( sqrt(7) ), t: 2)$] ⓑ #math.equation(block: false, alt: "open parenthesis − the square root of 15 close parenthesis squared")[$attach(( "−" sqrt(15) ), t: 2)$]

ⓐ #math.equation(block: false, alt: "open parenthesis the square root of 11 close parenthesis squared")[$attach(( sqrt(11) ), t: 2)$] ⓑ #math.equation(block: false, alt: "open parenthesis − the square root of 21 close parenthesis squared")[$attach(( "−" sqrt(21) ), t: 2)$]

#solutionbox[
ⓐ 11 ⓑ 21
]

ⓐ #math.equation(block: false, alt: "open parenthesis the square root of 19 close parenthesis squared")[$attach(( sqrt(19) ), t: 2)$] ⓑ #math.equation(block: false, alt: "open parenthesis − the square root of 5 close parenthesis squared")[$attach(( "−" sqrt(5) ), t: 2)$]

ⓐ #math.equation(block: false, alt: "open parenthesis the square root of 23 close parenthesis squared")[$attach(( sqrt(23) ), t: 2)$] #linebreak() ⓑ #math.equation(block: false, alt: "open parenthesis − the square root of 3 close parenthesis squared")[$attach(( "−" sqrt(3) ), t: 2)$]

#solutionbox[
ⓐ 23 ⓑ 3
]

ⓐ #math.equation(block: false, alt: "open parenthesis 4 the square root of 11 close parenthesis open parenthesis −3 the square root of 11 close parenthesis")[$( 4 sqrt(11) ) ( −3 sqrt(11) )$] ⓑ #math.equation(block: false, alt: "open parenthesis 5 the square root of 3 close parenthesis squared")[$attach(( 5 sqrt(3) ), t: 2)$]

ⓐ #math.equation(block: false, alt: "open parenthesis 2 the square root of 13 close parenthesis open parenthesis −9 the square root of 13 close parenthesis")[$( 2 sqrt(13) ) ( −9 sqrt(13) )$] ⓑ #math.equation(block: false, alt: "open parenthesis 6 the square root of 5 close parenthesis squared")[$attach(( 6 sqrt(5) ), t: 2)$]

#solutionbox[
ⓐ −234 ⓑ 180
]

ⓐ #math.equation(block: false, alt: "open parenthesis −3 the square root of 12 close parenthesis open parenthesis −2 the square root of 6 close parenthesis")[$( −3 sqrt(12) ) ( −2 sqrt(6) )$] ⓑ #math.equation(block: false, alt: "open parenthesis −4 the square root of 10 close parenthesis squared")[$attach(( −4 sqrt(10) ), t: 2)$]

ⓐ #math.equation(block: false, alt: "open parenthesis −7 the square root of 5 close parenthesis open parenthesis −3 the square root of 10 close parenthesis")[$( −7 sqrt(5) ) ( −3 sqrt(10) )$] ⓑ #math.equation(block: false, alt: "open parenthesis −2 the square root of 14 close parenthesis squared")[$attach(( −2 sqrt(14) ), t: 2)$]

#solutionbox[
ⓐ #math.equation(block: false, alt: "105 the square root of 2")[$105 sqrt(2)$] ⓑ 56
]

#strong[Use Polynomial Multiplication to Multiply Square Roots]

In the following exercises, simplify.

ⓐ #math.equation(block: false, alt: "3 open parenthesis 4 minus the square root of 3 close parenthesis")[$3 ( 4 − sqrt(3) )$] ⓑ #math.equation(block: false, alt: "the square root of 2 open parenthesis 4 minus the square root of 6 close parenthesis")[$sqrt(2) ( 4 − sqrt(6) )$]

ⓐ #math.equation(block: false, alt: "4 open parenthesis 6 minus the square root of 11 close parenthesis")[$4 ( 6 − sqrt(11) )$] ⓑ #math.equation(block: false, alt: "the square root of 2 open parenthesis 5 minus the square root of 12 close parenthesis")[$sqrt(2) ( 5 − sqrt(12) )$]

#solutionbox[
ⓐ #math.equation(block: false, alt: "24 minus 4 the square root of 11")[$24 − 4 sqrt(11)$] ⓑ #math.equation(block: false, alt: "5 the square root of 2 minus 2 the square root of 6")[$5 sqrt(2) − 2 sqrt(6)$]
]

ⓐ #math.equation(block: false, alt: "5 open parenthesis 3 minus the square root of 7 close parenthesis")[$5 ( 3 − sqrt(7) )$] ⓑ #math.equation(block: false, alt: "the square root of 3 open parenthesis 4 minus the square root of 15 close parenthesis")[$sqrt(3) ( 4 − sqrt(15) )$]

ⓐ #math.equation(block: false, alt: "7 open parenthesis −2 minus the square root of 11 close parenthesis")[$7 ( −2 − sqrt(11) )$] ⓑ #math.equation(block: false, alt: "the square root of 7 open parenthesis 6 minus the square root of 14 close parenthesis")[$sqrt(7) ( 6 − sqrt(14) )$]

#solutionbox[
ⓐ #math.equation(block: false, alt: "−14 minus 7 the square root of 11")[$−14 − 7 sqrt(11)$] ⓑ #math.equation(block: false, alt: "6 the square root of 7 minus 7 the square root of 2")[$6 sqrt(7) − 7 sqrt(2)$]
]

ⓐ #math.equation(block: false, alt: "the square root of 7 open parenthesis 5 plus 2 the square root of 7 close parenthesis")[$sqrt(7) ( 5 + 2 sqrt(7) )$] ⓑ #math.equation(block: false, alt: "the square root of 5 open parenthesis the square root of 10 plus the square root of 18 close parenthesis")[$sqrt(5) ( sqrt(10) + sqrt(18) )$]

ⓐ #math.equation(block: false, alt: "the square root of 11 open parenthesis 8 plus 4 the square root of 11 close parenthesis")[$sqrt(11) ( 8 + 4 sqrt(11) )$] ⓑ #math.equation(block: false, alt: "the square root of 3 open parenthesis the square root of 12 plus the square root of 27 close parenthesis")[$sqrt(3) ( sqrt(12) + sqrt(27) )$]

#solutionbox[
ⓐ #math.equation(block: false, alt: "44 plus 8 the square root of 11")[$44 + 8 sqrt(11)$] ⓑ #math.equation(block: false, alt: "15")[$15$]
]

ⓐ #math.equation(block: false, alt: "the square root of 11 open parenthesis −3 plus 4 the square root of 11 close parenthesis")[$sqrt(11) ( −3 + 4 sqrt(11) )$] ⓑ #math.equation(block: false, alt: "the square root of 3 open parenthesis the square root of 15 minus the square root of 18 close parenthesis")[$sqrt(3) ( sqrt(15) − sqrt(18) )$]

ⓐ #math.equation(block: false, alt: "the square root of 2 open parenthesis −5 plus 9 the square root of 2 close parenthesis")[$sqrt(2) ( −5 + 9 sqrt(2) )$] ⓑ #math.equation(block: false, alt: "the square root of 7 open parenthesis the square root of 3 minus the square root of 21 close parenthesis")[$sqrt(7) ( sqrt(3) − sqrt(21) )$]

#solutionbox[
ⓐ #math.equation(block: false, alt: "18 minus 5 the square root of 2")[$18 − 5 sqrt(2)$] ⓑ #math.equation(block: false, alt: "the square root of 21 minus 7 the square root of 3")[$sqrt(21) − 7 sqrt(3)$]
]

#math.equation(block: true, alt: "open parenthesis 8 plus the square root of 3 close parenthesis open parenthesis 2 minus the square root of 3 close parenthesis")[$( 8 + sqrt(3) ) ( 2 − sqrt(3) )$]

#math.equation(block: true, alt: "open parenthesis 7 plus the square root of 3 close parenthesis open parenthesis 9 minus the square root of 3 close parenthesis")[$( 7 + sqrt(3) ) ( 9 − sqrt(3) )$]

#solutionbox[
#math.equation(block: true, alt: "60 plus 2 the square root of 3")[$60 + 2 sqrt(3)$]
]

#math.equation(block: true, alt: "open parenthesis 8 minus the square root of 2 close parenthesis open parenthesis 3 plus the square root of 2 close parenthesis")[$( 8 − sqrt(2) ) ( 3 + sqrt(2) )$]

#math.equation(block: true, alt: "open parenthesis 9 minus the square root of 2 close parenthesis open parenthesis 6 plus the square root of 2 close parenthesis")[$( 9 − sqrt(2) ) ( 6 + sqrt(2) )$]

#solutionbox[
#math.equation(block: true, alt: "52 plus 3 the square root of 2")[$52 + 3 sqrt(2)$]
]

#math.equation(block: true, alt: "open parenthesis 3 minus the square root of 7 close parenthesis open parenthesis 5 minus the square root of 7 close parenthesis")[$( 3 − sqrt(7) ) ( 5 − sqrt(7) )$]

#math.equation(block: true, alt: "open parenthesis 5 minus the square root of 7 close parenthesis open parenthesis 4 minus the square root of 7 close parenthesis")[$( 5 − sqrt(7) ) ( 4 − sqrt(7) )$]

#solutionbox[
#math.equation(block: true, alt: "27 minus 9 the square root of 7")[$27 − 9 sqrt(7)$]
]

#math.equation(block: true, alt: "open parenthesis 1 plus 3 the square root of 10 close parenthesis open parenthesis 5 minus 2 the square root of 10 close parenthesis")[$( 1 + 3 sqrt(10) ) ( 5 − 2 sqrt(10) )$]

#math.equation(block: true, alt: "open parenthesis 7 minus 2 the square root of 5 close parenthesis open parenthesis 4 plus 9 the square root of 5 close parenthesis")[$( 7 − 2 sqrt(5) ) ( 4 + 9 sqrt(5) )$]

#solutionbox[
#math.equation(block: true, alt: "−62 plus 55 the square root of 5")[$−62 + 55 sqrt(5)$]
]

#math.equation(block: true, alt: "open parenthesis the square root of 3 plus the square root of 10 close parenthesis open parenthesis the square root of 3 plus 2 the square root of 10 close parenthesis")[$( sqrt(3) + sqrt(10) ) ( sqrt(3) + 2 sqrt(10) )$]

#math.equation(block: true, alt: "open parenthesis the square root of 11 plus the square root of 5 close parenthesis open parenthesis the square root of 11 plus 6 the square root of 5 close parenthesis")[$( sqrt(11) + sqrt(5) ) ( sqrt(11) + 6 sqrt(5) )$]

#solutionbox[
#math.equation(block: true, alt: "41 plus 7 the square root of 55")[$41 + 7 sqrt(55)$]
]

#math.equation(block: true, alt: "open parenthesis 2 the square root of 7 minus 5 the square root of 11 close parenthesis open parenthesis 4 the square root of 7 plus 9 the square root of 11 close parenthesis")[$( 2 sqrt(7) − 5 sqrt(11) ) ( 4 sqrt(7) + 9 sqrt(11) )$]

#math.equation(block: true, alt: "open parenthesis 4 the square root of 6 plus 7 the square root of 13 close parenthesis open parenthesis 8 the square root of 6 minus 3 the square root of 13 close parenthesis")[$( 4 sqrt(6) + 7 sqrt(13) ) ( 8 sqrt(6) − 3 sqrt(13) )$]

#solutionbox[
#math.equation(block: true, alt: "−81 plus 44 the square root of 78")[$−81 + 44 sqrt(78)$]
]

#math.equation(block: true, alt: "open parenthesis 5 minus the square root of u close parenthesis open parenthesis 3 plus the square root of u close parenthesis")[$( 5 − sqrt(u) ) ( 3 + sqrt(u) )$]

#math.equation(block: true, alt: "open parenthesis 9 minus the square root of w close parenthesis open parenthesis 2 plus the square root of w close parenthesis")[$( 9 − sqrt(w) ) ( 2 + sqrt(w) )$]

#solutionbox[
#math.equation(block: true, alt: "18 plus 7 the square root of w minus w")[$18 + 7 sqrt(w) − w$]
]

#math.equation(block: true, alt: "open parenthesis 7 plus 2 the square root of m close parenthesis open parenthesis 4 plus 9 the square root of m close parenthesis")[$( 7 + 2 sqrt(m) ) ( 4 + 9 sqrt(m) )$]

#math.equation(block: true, alt: "open parenthesis 6 plus 5 the square root of n close parenthesis open parenthesis 11 plus 3 the square root of n close parenthesis")[$( 6 + 5 sqrt(n) ) ( 11 + 3 sqrt(n) )$]

#solutionbox[
#math.equation(block: true, alt: "66 plus 73 the square root of n plus 15 n")[$66 + 73 sqrt(n) + 15 n$]
]

ⓐ #math.equation(block: false, alt: "open parenthesis 3 plus the square root of 5 close parenthesis squared")[$attach(( 3 + sqrt(5) ), t: 2)$] #linebreak() ⓑ #math.equation(block: false, alt: "open parenthesis 2 minus 5 the square root of 3 close parenthesis squared")[$attach(( 2 − 5 sqrt(3) ), t: 2)$]

ⓐ #math.equation(block: false, alt: "open parenthesis 4 plus the square root of 11 close parenthesis squared")[$attach(( 4 + sqrt(11) ), t: 2)$] ⓑ #math.equation(block: false, alt: "open parenthesis 3 minus 2 the square root of 5 close parenthesis squared")[$attach(( 3 − 2 sqrt(5) ), t: 2)$]

#solutionbox[
ⓐ #math.equation(block: false, alt: "27 plus 8 the square root of 11")[$27 + 8 sqrt(11)$] ⓑ #math.equation(block: false, alt: "29 minus 12 the square root of 5")[$29 − 12 sqrt(5)$]
]

ⓐ #math.equation(block: false, alt: "open parenthesis 9 minus the square root of 6 close parenthesis squared")[$attach(( 9 − sqrt(6) ), t: 2)$] ⓑ #math.equation(block: false, alt: "open parenthesis 10 plus 3 the square root of 7 close parenthesis squared")[$attach(( 10 + 3 sqrt(7) ), t: 2)$]

ⓐ #math.equation(block: false, alt: "open parenthesis 5 minus the square root of 10 close parenthesis squared")[$attach(( 5 − sqrt(10) ), t: 2)$] ⓑ #math.equation(block: false, alt: "open parenthesis 8 plus 3 the square root of 2 close parenthesis squared")[$attach(( 8 + 3 sqrt(2) ), t: 2)$]

#solutionbox[
ⓐ #math.equation(block: false, alt: "35 minus 10 the square root of 10")[$35 − 10 sqrt(10)$] ⓑ #math.equation(block: false, alt: "82 plus 48 the square root of 2")[$82 + 48 sqrt(2)$]
]

#math.equation(block: true, alt: "open parenthesis 3 minus the square root of 5 close parenthesis open parenthesis 3 plus the square root of 5 close parenthesis")[$( 3 − sqrt(5) ) ( 3 + sqrt(5) )$]

#math.equation(block: true, alt: "open parenthesis 10 minus the square root of 3 close parenthesis open parenthesis 10 plus the square root of 3 close parenthesis")[$( 10 − sqrt(3) ) ( 10 + sqrt(3) )$]

#solutionbox[
97
]

#math.equation(block: true, alt: "open parenthesis 4 plus the square root of 2 close parenthesis open parenthesis 4 minus the square root of 2 close parenthesis")[$( 4 + sqrt(2) ) ( 4 − sqrt(2) )$]

#math.equation(block: true, alt: "open parenthesis 7 plus the square root of 10 close parenthesis open parenthesis 7 minus the square root of 10 close parenthesis")[$( 7 + sqrt(10) ) ( 7 − sqrt(10) )$]

#solutionbox[
39
]

#math.equation(block: true, alt: "open parenthesis 4 plus 9 the square root of 3 close parenthesis open parenthesis 4 minus 9 the square root of 3 close parenthesis")[$( 4 + 9 sqrt(3) ) ( 4 − 9 sqrt(3) )$]

#math.equation(block: true, alt: "open parenthesis 1 plus 8 the square root of 2 close parenthesis open parenthesis 1 minus 8 the square root of 2 close parenthesis")[$( 1 + 8 sqrt(2) ) ( 1 − 8 sqrt(2) )$]

#solutionbox[
#math.equation(block: true, alt: "−127")[$−127$]
]

#math.equation(block: true, alt: "open parenthesis 12 minus 5 the square root of 5 close parenthesis open parenthesis 12 plus 5 the square root of 5 close parenthesis")[$( 12 − 5 sqrt(5) ) ( 12 + 5 sqrt(5) )$]

#math.equation(block: true, alt: "open parenthesis 9 minus 4 the square root of 3 close parenthesis open parenthesis 9 plus 4 the square root of 3 close parenthesis")[$( 9 − 4 sqrt(3) ) ( 9 + 4 sqrt(3) )$]

#solutionbox[
33
]

#strong[Mixed Practice]

In the following exercises, simplify.

#math.equation(block: true, alt: "the square root of 3 times the square root of 21")[$sqrt(3) · sqrt(21)$]

#math.equation(block: true, alt: "open parenthesis 4 the square root of 6 close parenthesis open parenthesis − the square root of 18 close parenthesis")[$( 4 sqrt(6) ) ( "−" sqrt(18) )$]

#solutionbox[
#math.equation(block: true, alt: "−24 the square root of 3")[$−24 sqrt(3)$]
]

#math.equation(block: true, alt: "open parenthesis −5 plus the square root of 7 close parenthesis open parenthesis 6 plus the square root of 21 close parenthesis")[$( −5 + sqrt(7) ) ( 6 + sqrt(21) )$]

#math.equation(block: true, alt: "open parenthesis −5 the square root of 7 close parenthesis open parenthesis 6 the square root of 21 close parenthesis")[$( −5 sqrt(7) ) ( 6 sqrt(21) )$]

#solutionbox[
#math.equation(block: true, alt: "−210 the square root of 3")[$−210 sqrt(3)$]
]

#math.equation(block: true, alt: "open parenthesis −4 the square root of 2 close parenthesis open parenthesis 2 the square root of 18 close parenthesis")[$( −4 sqrt(2) ) ( 2 sqrt(18) )$]

#math.equation(block: true, alt: "open parenthesis the square root of 35 y cubed close parenthesis open parenthesis the square root of 7 y cubed close parenthesis")[$( sqrt(35 y^(3)) ) ( sqrt(7 y^(3)) )$]

#solutionbox[
#math.equation(block: true, alt: "7 y cubed the square root of 5")[$7 y^(3) sqrt(5)$]
]

#math.equation(block: true, alt: "open parenthesis 4 the square root of 12 x to the power 5 close parenthesis open parenthesis 2 the square root of 6 x cubed close parenthesis")[$( 4 sqrt(12 x^(5)) ) ( 2 sqrt(6 x^(3)) )$]

#math.equation(block: true, alt: "open parenthesis the square root of 29 close parenthesis squared")[$attach(( sqrt(29) ), t: 2)$]

#solutionbox[
#math.equation(block: true, alt: "29")[$29$]
]

#math.equation(block: true, alt: "open parenthesis −4 the square root of 17 close parenthesis open parenthesis −3 the square root of 17 close parenthesis")[$( −4 sqrt(17) ) ( −3 sqrt(17) )$]

#math.equation(block: true, alt: "open parenthesis −4 plus the square root of 17 close parenthesis open parenthesis −3 plus the square root of 17 close parenthesis")[$( −4 + sqrt(17) ) ( −3 + sqrt(17) )$]

#solutionbox[
#math.equation(block: true, alt: "29 minus 7 the square root of 17")[$29 − 7 sqrt(17)$]
]


==== Everyday Math

A landscaper wants to put a square reflecting pool next to a triangular deck, as shown below. The triangular deck is a right triangle, with legs of length 9 feet and 11 feet, and the pool will be adjacent to the hypotenuse.

+ ⓐ Use the Pythagorean Theorem to find the length of a side of the pool. Round your answer to the nearest tenth of a foot.
+ ⓑ Find the exact area of the pool.

#figure(figph[This figure is an illustration of a square pool with a deck in the shape of a right triangle. the pool's sides are x inches long while the deck's hypotenuse is x inches long and its legs are nine and eleven inches long.], alt: "This figure is an illustration of a square pool with a deck in the shape of a right triangle. the pool's sides are x inches long while the deck's hypotenuse is x inches long and its legs are nine and eleven inches long.", caption: none)

An artist wants to make a small monument in the shape of a square base topped by a right triangle, as shown below. The square base will be adjacent to one leg of the triangle. The other leg of the triangle will measure 2 feet and the hypotenuse will be 5 feet.

+ ⓐ Use the Pythagorean Theorem to find the length of a side of the square base. Round your answer to the nearest tenth of a foot. #linebreak() 
#figure(figph[This figure shows a marble sculpture in the form of a square with a right triangle resting on top of it. The sides of the square are x inches long, the legs of the triangle are x and two inches long, and the hypotenuse of the triangle is five inches long.], alt: "This figure shows a marble sculpture in the form of a square with a right triangle resting on top of it. The sides of the square are x inches long, the legs of the triangle are x and two inches long, and the hypotenuse of the triangle is five inches long.", caption: none)
+ ⓑ Find the exact area of the face of the square base.

#solutionbox[
ⓐ #math.equation(block: false, alt: "4.6 feet")[$4.6 #h(0.2em) "feet"$] ⓑ #math.equation(block: false, alt: "21 sq. feet")[$21 #h(0.2em) "sq. feet"$]
]

A square garden will be made with a stone border on one edge. If only #math.equation(block: false, alt: "3 plus the square root of 10")[$3 + sqrt(10)$] feet of stone are available, simplify #math.equation(block: false, alt: "open parenthesis 3 plus the square root of 10 close parenthesis squared")[$attach(( 3 + sqrt(10) ), t: 2)$] to determine the area of the largest such garden. Round your answer to the nearest tenth of a foot.

A garden will be made so as to contain two square sections, one section with side length #math.equation(block: false, alt: "the square root of 5 plus the square root of 6")[$sqrt(5) + sqrt(6)$] yards and one section with side length #math.equation(block: false, alt: "the square root of 2 plus the square root of 3")[$sqrt(2) + sqrt(3)$] yards. Simplify #math.equation(block: false, alt: "the square root of 5 plus the square root of 6 squared plus the square root of 2 plus the square root of 3 squared")[$attach(sqrt(5) + sqrt(6), t: 2) + attach(sqrt(2) + sqrt(3), t: 2)$] to determine the total area of the garden. Round your answer to the nearest tenth.

#solutionbox[
#math.equation(block: true, alt: "31.9 square yards")[$"31.9" #h(0.2em) "square" #h(0.2em) "yards"$]
]

Suppose a third section will be added to the garden in the previous exercise. The third section is to have a width of #math.equation(block: false, alt: "the square root of 432")[$sqrt(432)$] yards. Write an expression that gives the total area of the garden.


==== Writing Exercises

+ ⓐ Explain why #math.equation(block: false, alt: "open parenthesis − the square root of n close parenthesis squared")[$attach(( "−" sqrt(n) ), t: 2)$] is always positive, for #math.equation(block: false, alt: "n greater than or equal to 0")[$n ≥ 0$].
+ ⓑ Explain why #math.equation(block: false, alt: "minus the square root of n squared")[$− sqrt(n)^(2)$] is always negative, for #math.equation(block: false, alt: "n greater than or equal to 0")[$n ≥ 0$].

#solutionbox[
ⓐ when squaring a negative, it becomes a positive ⓑ since the negative is not included in the parenthesis, it is not squared, and remains negative
]

Use the binomial square pattern to simplify #math.equation(block: false, alt: "open parenthesis 3 plus the square root of 2 close parenthesis squared")[$attach(( 3 + sqrt(2) ), t: 2)$]. Explain all your steps.


==== Self Check

ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.

#figure(figph[This table has four columns and three rows. The columns are labeled, “I can…,” “confidently.,” “with some help.,” and “no minus I don’t get it!” The rows under the “I can…” column read, “multiply square roots.,” and “use polynomial multiplication to multiply square roots.” The other rows under the other columns are empty.], alt: "This table has four columns and three rows. The columns are labeled, “I can…,” “confidently.,” “with some help.,” and “no minus I don’t get it!” The rows under the “I can…” column read, “multiply square roots.,” and “use polynomial multiplication to multiply square roots.” The other rows under the other columns are empty.", caption: none)

ⓑ On a scale of 1–10, how would you rate your mastery of this section in light of your responses on the checklist? How can you improve this?