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== 9.3#h(0.6em)Add and Subtract Square Roots

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

Add: ⓐ #math.equation(block: false, alt: "3 x plus 9 x")[$3 x + 9 x$] ⓑ #math.equation(block: false, alt: "5 m plus 5 n")[$5 m + 5 n$]. #linebreak() 
If you missed this problem, review .

#solutionbox[
ⓐ #math.equation(block: false, alt: "12 x")[$12 x$]
ⓑ #math.equation(block: false, alt: "5 m plus 5 n")[$5 m + 5 n$]
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Simplify: #math.equation(block: false, alt: "the square root of 50 x cubed")[$sqrt(50 x^(3))$]. #linebreak() 
If you missed this problem, review .

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

We know that we must follow the order of operations to simplify expressions with square roots. The radical is a grouping symbol, so we work inside the radical first. We simplify #math.equation(block: false, alt: "the square root of 2 plus 7")[$sqrt(2 + 7)$] in this way:

#math.equation(block: true, alt: "the square root of 2 plus 7; Add inside the radical., the square root of 9; Simplify., 3")[$&  &  & sqrt(2 + 7) \ "Add inside the radical." &  &  & sqrt(9) \ "Simplify." &  &  & 3$]

So if we have to add #math.equation(block: false, alt: "the square root of 2 plus the square root of 7")[$sqrt(2) + sqrt(7)$], we must not combine them into one radical.

#math.equation(block: true, alt: "the square root of 2 plus the square root of 7 not equal to the square root of 2 plus 7")[$sqrt(2) + sqrt(7) ≠ sqrt(2 + 7)$]Trying to add square roots with different radicands is like trying to add unlike terms.

#math.equation(block: true, alt: "But, just like we can add, x plus x ,, we can add, the square root of 3 plus the square root of 3 .; x plus x equals 2 x, the square root of 3 plus the square root of 3 equals 2 the square root of 3")[$"But, just like we can add" & x + x , & "we can add" & sqrt(3) + sqrt(3) . \  & x + x = 2 x &  & sqrt(3) + sqrt(3) = 2 sqrt(3)$]

Adding square roots with the same radicand is just like adding like terms. We call square roots with the same radicand like square roots to remind us they work the same as like terms.

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

Square roots with the same radicand are called #strong[like square roots].
]

We add and subtract like square roots in the same way we add and subtract like terms. We know that #math.equation(block: false, alt: "3 x plus 8 x")[$3 x + 8 x$] is #math.equation(block: false, alt: "11 x")[$11 x$]. Similarly we add #math.equation(block: false, alt: "3 the square root of x plus 8 the square root of x")[$3 sqrt(x) + 8 sqrt(x)$] and the result is #math.equation(block: false, alt: "11 the square root of x .")[$11 sqrt(x) .$]


=== Add and Subtract Like Square Roots

Think about adding like terms with variables as you do the next few examples. When you have like radicands, you just add or subtract the coefficients. When the radicands are not like, you cannot combine the terms.

#examplebox("Example 1")[][
Simplify: #math.equation(block: false, alt: "2 the square root of 2 minus 7 the square root of 2")[$2 sqrt(2) − 7 sqrt(2)$].

#solutionbox[
#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([], [#math.equation(block: false, alt: "2 the square root of 2 minus 7 the square root of 2")[$2 sqrt(2) − 7 sqrt(2)$]]),
  [Since the radicals are like, we subtract the coefficients.], [#math.equation(block: false, alt: "−5 the square root of 2")[$−5 sqrt(2)$]],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Simplify: #math.equation(block: false, alt: "8 the square root of 2 minus 9 the square root of 2")[$8 sqrt(2) − 9 sqrt(2)$].

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

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Simplify: #math.equation(block: false, alt: "5 the square root of 3 minus 9 the square root of 3")[$5 sqrt(3) − 9 sqrt(3)$].

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

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

#solutionbox[
#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([], [#math.equation(block: false, alt: "3 the square root of y plus 4 the square root of y")[$3 sqrt(y) + 4 sqrt(y)$]]),
  [Since the radicals are like, we add the coefficients.], [#math.equation(block: false, alt: "7 the square root of y")[$7 sqrt(y)$]],
))
]
]

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

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

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

#solutionbox[
#math.equation(block: true, alt: "8 the square root of u")[$8 sqrt(u)$]
]
]

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

#solutionbox[
#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([], [#math.equation(block: false, alt: "4 the square root of x minus 2 the square root of y")[$4 sqrt(x) − 2 sqrt(y)$]]),
  [Since the radicals are not like, we cannot #linebreak() subtract them. We leave the expression as is.], [#math.equation(block: false, alt: "4 the square root of x minus 2 the square root of y")[$4 sqrt(x) − 2 sqrt(y)$]],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Simplify: #math.equation(block: false, alt: "7 the square root of p minus 6 the square root of q")[$7 sqrt(p) − 6 sqrt(q)$].

#solutionbox[
#math.equation(block: true, alt: "7 the square root of p minus 6 the square root of q")[$7 sqrt(p) − 6 sqrt(q)$]
]
]

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

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

#examplebox("Example 4")[][
Simplify: #math.equation(block: false, alt: "5 the square root of 13 plus 4 the square root of 13 plus 2 the square root of 13")[$5 sqrt(13) + 4 sqrt(13) + 2 sqrt(13)$].

#solutionbox[
#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([], [#math.equation(block: false, alt: "5 the square root of 13 plus 4 the square root of 13 plus 2 the square root of 13")[$5 sqrt(13) + 4 sqrt(13) + 2 sqrt(13)$]]),
  [Since the radicals are like, we add the coefficients.], [#math.equation(block: false, alt: "11 the square root of 13")[$11 sqrt(13)$]],
))
]
]

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

#solutionbox[
#math.equation(block: true, alt: "9 the square root of 11")[$9 sqrt(11)$]
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Simplify: #math.equation(block: false, alt: "6 the square root of 10 plus 2 the square root of 10 plus 3 the square root of 10")[$6 sqrt(10) + 2 sqrt(10) + 3 sqrt(10)$].

#solutionbox[
#math.equation(block: true, alt: "11 the square root of 10")[$11 sqrt(10)$]
]
]

#examplebox("Example 5")[][
Simplify: #math.equation(block: false, alt: "2 the square root of 6 minus 6 the square root of 6 plus 3 the square root of 3")[$2 sqrt(6) − 6 sqrt(6) + 3 sqrt(3)$].

#solutionbox[
#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([], [#math.equation(block: false, alt: "2 the square root of 6 minus 6 the square root of 6 plus 3 the square root of 3")[$2 sqrt(6) − 6 sqrt(6) + 3 sqrt(3)$]]),
  [Since the first two radicals are like, we #linebreak() subtract their coefficients.], [#math.equation(block: false, alt: "minus 4 the square root of 6 plus 3 the square root of 3")[$− 4 sqrt(6) + 3 sqrt(3)$]],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Simplify: #math.equation(block: false, alt: "5 the square root of 5 minus 4 the square root of 5 plus 2 the square root of 6")[$5 sqrt(5) − 4 sqrt(5) + 2 sqrt(6)$].

#solutionbox[
#math.equation(block: true, alt: "the square root of 5 plus 2 the square root of 6")[$sqrt(5) + 2 sqrt(6)$]
]
]

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

#solutionbox[
#math.equation(block: true, alt: "−5 the square root of 7 plus 2 the square root of 5")[$−5 sqrt(7) + 2 sqrt(5)$]
]
]

#examplebox("Example 6")[][
Simplify: #math.equation(block: false, alt: "2 the square root of 5 n minus 6 the square root of 5 n plus 4 the square root of 5 n")[$2 sqrt(5 n) − 6 sqrt(5 n) + 4 sqrt(5 n)$].

#solutionbox[
#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([], [#math.equation(block: false, alt: "2 the square root of 5 n minus 6 the square root of 5 n plus 4 the square root of 5 n")[$2 sqrt(5 n) − 6 sqrt(5 n) + 4 sqrt(5 n)$]]),
  [Since the radicals are like, we combine them.], [#math.equation(block: false, alt: "0 the square root of 5 n")[$0 sqrt(5 n)$]],
  [Simplify.], [0],
))
]
]

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

#solutionbox[
#math.equation(block: true, alt: "−2 the square root of 7 x")[$−2 sqrt(7 x)$]
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Simplify: #math.equation(block: false, alt: "4 the square root of 3 y minus 7 the square root of 3 y plus 2 the square root of 3 y")[$4 sqrt(3 y) − 7 sqrt(3 y) + 2 sqrt(3 y)$].

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

When radicals contain more than one variable, as long as all the variables and their exponents are identical, the radicals are like.

#examplebox("Example 7")[][
Simplify: #math.equation(block: false, alt: "the square root of 3 x y plus 5 the square root of 3 x y minus 4 the square root of 3 x y")[$sqrt(3 x y) + 5 sqrt(3 x y) − 4 sqrt(3 x y)$].

#solutionbox[
#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([], [#math.equation(block: false, alt: "the square root of 3 x y plus 5 the square root of 3 x y minus 4 the square root of 3 x y")[$sqrt(3 x y) + 5 sqrt(3 x y) − 4 sqrt(3 x y)$]]),
  [Since the radicals are like, we combine them.], [#math.equation(block: false, alt: "2 the square root of 3 x y")[$2 sqrt(3 x y)$]],
))
]
]

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

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

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

#solutionbox[
#math.equation(block: true, alt: "0")[$0$]
]
]


=== Add and Subtract Square Roots that Need Simplification

Remember that we always simplify square roots by removing the largest perfect-square factor. Sometimes when we have to add or subtract square roots that do not appear to have #strong[like radicals], we find like radicals after simplifying the square roots.

#examplebox("Example 8")[][
Simplify: #math.equation(block: false, alt: "the square root of 20 plus 3 the square root of 5")[$sqrt(20) + 3 sqrt(5)$].

#solutionbox[
#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([], [#math.equation(block: false, alt: "the square root of 20 plus 3 the square root of 5")[$sqrt(20) + 3 sqrt(5)$]]),
  [Simplify the radicals, when possible.], [#math.equation(block: false, alt: "the square root of 4 times the square root of 5 plus 3 the square root of 5")[$sqrt(4) · sqrt(5) + 3 sqrt(5)$]],
  [], [#math.equation(block: false, alt: "2 the square root of 5 plus 3 the square root of 5")[$2 sqrt(5) + 3 sqrt(5)$]],
  [Combine the like radicals.], [#math.equation(block: false, alt: "5 the square root of 5")[$5 sqrt(5)$]],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Simplify: #math.equation(block: false, alt: "the square root of 18 plus 6 the square root of 2")[$sqrt(18) + 6 sqrt(2)$].

#solutionbox[
#math.equation(block: true, alt: "9 the square root of 2")[$9 sqrt(2)$]
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Simplify: #math.equation(block: false, alt: "the square root of 27 plus 4 the square root of 3")[$sqrt(27) + 4 sqrt(3)$].

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

#examplebox("Example 9")[][
Simplify: #math.equation(block: false, alt: "the square root of 48 minus the square root of 75")[$sqrt(48) − sqrt(75)$].

#solutionbox[
#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([], [#math.equation(block: false, alt: "the square root of 48 minus the square root of 75")[$sqrt(48) − sqrt(75)$]]),
  [Simplify the radicals.], [#math.equation(block: false, alt: "the square root of 16 times the square root of 3 minus the square root of 25 times the square root of 3")[$sqrt(16) · sqrt(3) − sqrt(25) · sqrt(3)$]],
  [], [#math.equation(block: false, alt: "4 the square root of 3 minus 5 the square root of 3")[$4 sqrt(3) − 5 sqrt(3)$]],
  [Combine the like radicals.], [#math.equation(block: false, alt: "− the square root of 3")[$"−" sqrt(3)$]],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Simplify: #math.equation(block: false, alt: "the square root of 32 minus the square root of 18")[$sqrt(32) − sqrt(18)$].

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

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Simplify: #math.equation(block: false, alt: "the square root of 20 minus the square root of 45")[$sqrt(20) − sqrt(45)$].

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

Just like we use the Associative Property of Multiplication to simplify #math.equation(block: false, alt: "5 open parenthesis 3 x close parenthesis")[$5 ( 3 x )$] and get #math.equation(block: false, alt: "15 x")[$15 x$], we can simplify #math.equation(block: false, alt: "5 open parenthesis 3 the square root of x close parenthesis")[$5 ( 3 sqrt(x) )$] and get #math.equation(block: false, alt: "15 the square root of x")[$15 sqrt(x)$]. We will use the Associative Property to do this in the next example.

#examplebox("Example 10")[][
Simplify: #math.equation(block: false, alt: "5 the square root of 18 minus 2 the square root of 8")[$5 sqrt(18) − 2 sqrt(8)$].

#solutionbox[
#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([], [#math.equation(block: false, alt: "5 the square root of 18 minus 2 the square root of 8")[$5 sqrt(18) − 2 sqrt(8)$]]),
  [Simplify the radicals.], [#math.equation(block: false, alt: "5 times the square root of 9 times the square root of 2 minus 2 times the square root of 4 times the square root of 2")[$5 · sqrt(9) · sqrt(2) − 2 · sqrt(4) · sqrt(2)$]],
  [], [#math.equation(block: false, alt: "5 times 3 times the square root of 2 minus 2 times 2 times the square root of 2")[$5 · 3 · sqrt(2) − 2 · 2 · sqrt(2)$]],
  [], [#math.equation(block: false, alt: "15 the square root of 2 minus 4 the square root of 2")[$15 sqrt(2) − 4 sqrt(2)$]],
  [Combine the like radicals.], [#math.equation(block: false, alt: "11 the square root of 2")[$11 sqrt(2)$]],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Simplify: #math.equation(block: false, alt: "4 the square root of 27 minus 3 the square root of 12")[$4 sqrt(27) − 3 sqrt(12)$].

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

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Simplify: #math.equation(block: false, alt: "3 the square root of 20 minus 7 the square root of 45")[$3 sqrt(20) − 7 sqrt(45)$].

#solutionbox[
#math.equation(block: true, alt: "−15 the square root of 5")[$−15 sqrt(5)$]
]
]

#examplebox("Example 11")[][
Simplify: #math.equation(block: false, alt: "the fraction 3 over 4 the square root of 192 minus the fraction 5 over 6 the square root of 108")[$frac(3, 4) sqrt(192) − frac(5, 6) sqrt(108)$].

#solutionbox[
#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([], [#math.equation(block: false, alt: "the fraction 3 over 4 the square root of 192 minus the fraction 5 over 6 the square root of 108")[$frac(3, 4) sqrt(192) − frac(5, 6) sqrt(108)$]]),
  [Simplify the radicals.], [#math.equation(block: false, alt: "the fraction 3 over 4 the square root of 64 times the square root of 3 minus the fraction 5 over 6 the square root of 36 times the square root of 3")[$frac(3, 4) sqrt(64) · sqrt(3) − frac(5, 6) sqrt(36) · sqrt(3)$]],
  [], [#math.equation(block: false, alt: "the fraction 3 over 4 times 8 times the square root of 3 minus the fraction 5 over 6 times 6 times the square root of 3")[$frac(3, 4) · 8 · sqrt(3) − frac(5, 6) · 6 · sqrt(3)$]],
  [], [#math.equation(block: false, alt: "6 the square root of 3 minus 5 the square root of 3")[$6 sqrt(3) − 5 sqrt(3)$]],
  [Combine the like radicals.], [#math.equation(block: false, alt: "the square root of 3")[$sqrt(3)$]],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Simplify: #math.equation(block: false, alt: "the fraction 2 over 3 the square root of 108 minus the fraction 5 over 7 the square root of 147")[$frac(2, 3) sqrt(108) − frac(5, 7) sqrt(147)$].

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

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Simplify: #math.equation(block: false, alt: "the fraction 3 over 5 the square root of 200 minus the fraction 3 over 4 the square root of 128")[$frac(3, 5) sqrt(200) − frac(3, 4) sqrt(128)$].

#solutionbox[
#math.equation(block: true, alt: "0")[$0$]
]
]

#examplebox("Example 12")[][
Simplify: #math.equation(block: false, alt: "the fraction 2 over 3 the square root of 48 minus the fraction 3 over 4 the square root of 12")[$frac(2, 3) sqrt(48) − frac(3, 4) sqrt(12)$].

#solutionbox[
#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([], [#math.equation(block: false, alt: "the fraction 2 over 3 the square root of 48 minus the fraction 3 over 4 the square root of 12")[$frac(2, 3) sqrt(48) − frac(3, 4) sqrt(12)$]]),
  [Simplify the radicals.], [#math.equation(block: false, alt: "the fraction 2 over 3 the square root of 16 times the square root of 3 minus the fraction 3 over 4 the square root of 4 times the square root of 3")[$frac(2, 3) sqrt(16) · sqrt(3) − frac(3, 4) sqrt(4) · sqrt(3)$]],
  [], [#math.equation(block: false, alt: "the fraction 2 over 3 times 4 times the square root of 3 minus the fraction 3 over 4 times 2 times the square root of 3")[$frac(2, 3) · 4 · sqrt(3) − frac(3, 4) · 2 · sqrt(3)$]],
  [], [#math.equation(block: false, alt: "the fraction 8 over 3 the square root of 3 minus the fraction 3 over 2 the square root of 3")[$frac(8, 3) sqrt(3) − frac(3, 2) sqrt(3)$]],
  [Find a common denominator to subtract the #linebreak() coefficients of the like radicals.], [#math.equation(block: false, alt: "the fraction 16 over 6 the square root of 3 minus the fraction 9 over 6 the square root of 3")[$frac(16, 6) sqrt(3) − frac(9, 6) sqrt(3)$]],
  [Simplify.], [#math.equation(block: false, alt: "the fraction 7 over 6 the square root of 3")[$frac(7, 6) sqrt(3)$]],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Simplify: #math.equation(block: false, alt: "the fraction 2 over 5 the square root of 32 minus the fraction 1 over 3 the square root of 8")[$frac(2, 5) sqrt(32) − frac(1, 3) sqrt(8)$].

#solutionbox[
#math.equation(block: true, alt: "the fraction 14 over 15 the square root of 2")[$frac(14, 15) sqrt(2)$]
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Simplify: #math.equation(block: false, alt: "the fraction 1 over 3 the square root of 80 minus the fraction 1 over 4 the square root of 125")[$frac(1, 3) sqrt(80) − frac(1, 4) sqrt(125)$].

#solutionbox[
#math.equation(block: true, alt: "the fraction 1 over 12 the square root of 5")[$frac(1, 12) sqrt(5)$]
]
]

In the next example, we will remove constant and variable factors from the square roots.

#examplebox("Example 13")[][
Simplify: #math.equation(block: false, alt: "the square root of 18 n to the power 5 minus the square root of 32 n to the power 5")[$sqrt(18 n^(5)) − sqrt(32 n^(5))$].

#solutionbox[
#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([], [#math.equation(block: false, alt: "the square root of 18 n to the power 5 minus the square root of 32 n to the power 5")[$sqrt(18 n^(5)) − sqrt(32 n^(5))$]]),
  [Simplify the radicals.], [#math.equation(block: false, alt: "the square root of 9 n to the power 4 times the square root of 2 n minus the square root of 16 n to the power 4 times the square root of 2 n")[$sqrt(9 n^(4)) · sqrt(2 n) − sqrt(16 n^(4)) · sqrt(2 n)$]],
  [], [#math.equation(block: false, alt: "3 n squared the square root of 2 n minus 4 n squared the square root of 2 n")[$3 n^(2) sqrt(2 n) − 4 n^(2) sqrt(2 n)$]],
  [Combine the like radicals.], [#math.equation(block: false, alt: "− n squared the square root of 2 n")[$"−" n^(2) sqrt(2 n)$]],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Simplify: #math.equation(block: false, alt: "the square root of 32 m to the power 7 minus the square root of 50 m to the power 7")[$sqrt(32 m^(7)) − sqrt(50 m^(7))$].

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

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Simplify: #math.equation(block: false, alt: "the square root of 27 p cubed minus the square root of 48 p cubed")[$sqrt(27 p^(3)) − sqrt(48 p^(3))$].

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

#examplebox("Example 14")[][
Simplify: #math.equation(block: false, alt: "9 the square root of 50 m squared minus 6 the square root of 48 m squared")[$9 sqrt(50 m^(2)) − 6 sqrt(48 m^(2))$].

#solutionbox[
#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([], [#math.equation(block: false, alt: "9 the square root of 50 m squared minus 6 the square root of 48 m squared")[$9 sqrt(50 m^(2)) − 6 sqrt(48 m^(2))$]]),
  [Simplify the radicals.], [#math.equation(block: false, alt: "9 the square root of 25 m squared times the square root of 2 minus 6 the square root of 16 m squared times the square root of 3")[$9 sqrt(25 m^(2)) · sqrt(2) − 6 sqrt(16 m^(2)) · sqrt(3)$]],
  [], [#math.equation(block: false, alt: "9 times 5 m times the square root of 2 minus 6 times 4 m times the square root of 3")[$9 · 5 m · sqrt(2) − 6 · 4 m · sqrt(3)$]],
  [], [#math.equation(block: false, alt: "45 m the square root of 2 minus 24 m the square root of 3")[$45 m sqrt(2) − 24 m sqrt(3)$]],
  [The radicals are not like and so cannot be combined.], [],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Simplify: #math.equation(block: false, alt: "5 the square root of 32 x squared minus 3 the square root of 48 x squared")[$5 sqrt(32 x^(2)) − 3 sqrt(48 x^(2))$].

#solutionbox[
#math.equation(block: true, alt: "20 x the square root of 2 minus 12 x the square root of 3")[$20 x sqrt(2) − 12 x sqrt(3)$]
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Simplify: #math.equation(block: false, alt: "7 the square root of 48 y squared minus 4 the square root of 72 y squared")[$7 sqrt(48 y^(2)) − 4 sqrt(72 y^(2))$].

#solutionbox[
#math.equation(block: true, alt: "28 y the square root of 3 minus 24 y the square root of 2")[$28 y sqrt(3) − 24 y sqrt(2)$]
]
]

#examplebox("Example 15")[][
Simplify: #math.equation(block: false, alt: "2 the square root of 8 x squared minus 5 x the square root of 32 plus 5 the square root of 18 x squared")[$2 sqrt(8 x^(2)) − 5 x sqrt(32) + 5 sqrt(18 x^(2))$].

#solutionbox[
#figure(table(
  columns: 2, align: left, inset: 6pt,
  table.header([], [#math.equation(block: false, alt: "2 the square root of 8 x squared minus 5 x the square root of 32 plus 5 the square root of 18 x squared")[$2 sqrt(8 x^(2)) − 5 x sqrt(32) + 5 sqrt(18 x^(2))$]]),
  [Simplify the radicals.], [#math.equation(block: false, alt: "2 the square root of 4 x squared times the square root of 2 minus 5 x the square root of 16 times the square root of 2 plus 5 the square root of 9 x squared times the square root of 2")[$2 sqrt(4 x^(2)) · sqrt(2) − 5 x sqrt(16) · sqrt(2) + 5 sqrt(9 x^(2)) · sqrt(2)$]],
  [], [#math.equation(block: false, alt: "2 times 2 x times the square root of 2 minus 5 x times 4 times the square root of 2 plus 5 times 3 x times the square root of 2")[$2 · 2 x · sqrt(2) − 5 x · 4 · sqrt(2) + 5 · 3 x · sqrt(2)$]],
  [], [#math.equation(block: false, alt: "4 x the square root of 2 minus 20 x the square root of 2 plus 15 x the square root of 2")[$4 x sqrt(2) − 20 x sqrt(2) + 15 x sqrt(2)$]],
  [Combine the like radicals.], [#math.equation(block: false, alt: "− x the square root of 2")[$"−" x sqrt(2)$]],
))
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Simplify: #math.equation(block: false, alt: "3 the square root of 12 x squared minus 2 x the square root of 48 plus 4 the square root of 27 x squared")[$3 sqrt(12 x^(2)) − 2 x sqrt(48) + 4 sqrt(27 x^(2))$].

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

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Simplify: #math.equation(block: false, alt: "3 the square root of 18 x squared minus 6 x the square root of 32 plus 2 the square root of 50 x squared")[$3 sqrt(18 x^(2)) − 6 x sqrt(32) + 2 sqrt(50 x^(2))$].

#solutionbox[
#math.equation(block: true, alt: "−5 x the square root of 2")[$−5 x sqrt(2)$]
]
]

#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Access this online resource for additional instruction and practice with the adding and subtracting square roots.

- #link("https://openstax.org/l/25AddSubtrSR")[Adding/Subtracting Square Roots]
]


=== Key Concepts

- To add or subtract like square roots, add or subtract the coefficients and keep the like square root.
- Sometimes when we have to add or subtract square roots that do not appear to have like radicals, we find like radicals after simplifying the square roots.


==== Practice Makes Perfect

#strong[Add and Subtract Like Square Roots]

In the following exercises, simplify.

#math.equation(block: true, alt: "8 the square root of 2 minus 5 the square root of 2")[$8 sqrt(2) − 5 sqrt(2)$]

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

#math.equation(block: true, alt: "7 the square root of 2 minus 3 the square root of 2")[$7 sqrt(2) − 3 sqrt(2)$]

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

#solutionbox[
#math.equation(block: true, alt: "9 the square root of 5")[$9 sqrt(5)$]
]

#math.equation(block: true, alt: "4 the square root of 5 plus 8 the square root of 5")[$4 sqrt(5) + 8 sqrt(5)$]

#math.equation(block: true, alt: "9 the square root of 7 minus 10 the square root of 7")[$9 sqrt(7) − 10 sqrt(7)$]

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

#math.equation(block: true, alt: "11 the square root of 7 minus 12 the square root of 7")[$11 sqrt(7) − 12 sqrt(7)$]

#math.equation(block: true, alt: "7 the square root of y plus 2 the square root of y")[$7 sqrt(y) + 2 sqrt(y)$]

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

#math.equation(block: true, alt: "9 the square root of n plus 3 the square root of n")[$9 sqrt(n) + 3 sqrt(n)$]

#math.equation(block: true, alt: "the square root of a minus 4 the square root of a")[$sqrt(a) − 4 sqrt(a)$]

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

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

#math.equation(block: true, alt: "5 the square root of c plus 2 the square root of c")[$5 sqrt(c) + 2 sqrt(c)$]

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

#math.equation(block: true, alt: "7 the square root of d plus 2 the square root of d")[$7 sqrt(d) + 2 sqrt(d)$]

#math.equation(block: true, alt: "8 the square root of a minus 2 the square root of b")[$8 sqrt(a) − 2 sqrt(b)$]

#solutionbox[
#math.equation(block: true, alt: "8 the square root of a minus 2 the square root of b")[$8 sqrt(a) − 2 sqrt(b)$]
]

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

#math.equation(block: true, alt: "5 the square root of m plus the square root of n")[$5 sqrt(m) + sqrt(n)$]

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

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

#math.equation(block: true, alt: "8 the square root of 7 plus 2 the square root of 7 plus 3 the square root of 7")[$8 sqrt(7) + 2 sqrt(7) + 3 sqrt(7)$]

#solutionbox[
#math.equation(block: true, alt: "13 the square root of 7")[$13 sqrt(7)$]
]

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

#math.equation(block: true, alt: "3 the square root of 11 plus 2 the square root of 11 minus 8 the square root of 11")[$3 sqrt(11) + 2 sqrt(11) − 8 sqrt(11)$]

#solutionbox[
#math.equation(block: true, alt: "−3 the square root of 11")[$−3 sqrt(11)$]
]

#math.equation(block: true, alt: "2 the square root of 15 plus 5 the square root of 15 minus 9 the square root of 15")[$2 sqrt(15) + 5 sqrt(15) − 9 sqrt(15)$]

#math.equation(block: true, alt: "3 the square root of 3 minus 8 the square root of 3 plus 7 the square root of 5")[$3 sqrt(3) − 8 sqrt(3) + 7 sqrt(5)$]

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

#math.equation(block: true, alt: "5 the square root of 7 minus 8 the square root of 7 plus 6 the square root of 3")[$5 sqrt(7) − 8 sqrt(7) + 6 sqrt(3)$]

#math.equation(block: true, alt: "6 the square root of 2 plus 2 the square root of 2 minus 3 the square root of 5")[$6 sqrt(2) + 2 sqrt(2) − 3 sqrt(5)$]

#solutionbox[
#math.equation(block: true, alt: "8 the square root of 2 minus 3 the square root of 5")[$8 sqrt(2) − 3 sqrt(5)$]
]

#math.equation(block: true, alt: "7 the square root of 5 plus the square root of 5 minus 8 the square root of 10")[$7 sqrt(5) + sqrt(5) − 8 sqrt(10)$]

#math.equation(block: true, alt: "3 the square root of 2 a minus 4 the square root of 2 a plus 5 the square root of 2 a")[$3 sqrt(2 a) − 4 sqrt(2 a) + 5 sqrt(2 a)$]

#solutionbox[
#math.equation(block: true, alt: "4 the square root of 2 a")[$4 sqrt(2 a)$]
]

#math.equation(block: true, alt: "the square root of 11 b minus 5 the square root of 11 b plus 3 the square root of 11 b")[$sqrt(11 b) − 5 sqrt(11 b) + 3 sqrt(11 b)$]

#math.equation(block: true, alt: "8 the square root of 3 c plus 2 the square root of 3 c minus 9 the square root of 3 c")[$8 sqrt(3 c) + 2 sqrt(3 c) − 9 sqrt(3 c)$]

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

#math.equation(block: true, alt: "3 the square root of 5 d plus 8 the square root of 5 d minus 11 the square root of 5 d")[$3 sqrt(5 d) + 8 sqrt(5 d) − 11 sqrt(5 d)$]

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

#solutionbox[
#math.equation(block: true, alt: "4 the square root of 3 a b")[$4 sqrt(3 a b)$]
]

#math.equation(block: true, alt: "8 the square root of 11 c d plus 5 the square root of 11 c d minus 9 the square root of 11 c d")[$8 sqrt(11 c d) + 5 sqrt(11 c d) − 9 sqrt(11 c d)$]

#math.equation(block: true, alt: "2 the square root of p q minus 5 the square root of p q plus 4 the square root of p q")[$2 sqrt(p q) − 5 sqrt(p q) + 4 sqrt(p q)$]

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

#math.equation(block: true, alt: "11 the square root of 2 r s minus 9 the square root of 2 r s plus 3 the square root of 2 r s")[$11 sqrt(2 r s) − 9 sqrt(2 r s) + 3 sqrt(2 r s)$]

#strong[Add and Subtract Square Roots that Need Simplification]

In the following exercises, simplify.

#math.equation(block: true, alt: "the square root of 50 plus 4 the square root of 2")[$sqrt(50) + 4 sqrt(2)$]

#solutionbox[
#math.equation(block: true, alt: "9 the square root of 2")[$9 sqrt(2)$]
]

#math.equation(block: true, alt: "the square root of 48 plus 2 the square root of 3")[$sqrt(48) + 2 sqrt(3)$]

#math.equation(block: true, alt: "the square root of 80 minus 3 the square root of 5")[$sqrt(80) − 3 sqrt(5)$]

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

#math.equation(block: true, alt: "the square root of 28 minus 4 the square root of 7")[$sqrt(28) − 4 sqrt(7)$]

#math.equation(block: true, alt: "the square root of 27 minus the square root of 75")[$sqrt(27) − sqrt(75)$]

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

#math.equation(block: true, alt: "the square root of 72 minus the square root of 98")[$sqrt(72) − sqrt(98)$]

#math.equation(block: true, alt: "the square root of 48 plus the square root of 27")[$sqrt(48) + sqrt(27)$]

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

#math.equation(block: true, alt: "the square root of 45 plus the square root of 80")[$sqrt(45) + sqrt(80)$]

#math.equation(block: true, alt: "2 the square root of 50 minus 3 the square root of 72")[$2 sqrt(50) − 3 sqrt(72)$]

#solutionbox[
#math.equation(block: true, alt: "−8 the square root of 2")[$−8 sqrt(2)$]
]

#math.equation(block: true, alt: "3 the square root of 98 minus the square root of 128")[$3 sqrt(98) − sqrt(128)$]

#math.equation(block: true, alt: "2 the square root of 12 plus 3 the square root of 48")[$2 sqrt(12) + 3 sqrt(48)$]

#solutionbox[
#math.equation(block: true, alt: "16 the square root of 3")[$16 sqrt(3)$]
]

#math.equation(block: true, alt: "4 the square root of 75 plus 2 the square root of 108")[$4 sqrt(75) + 2 sqrt(108)$]

#math.equation(block: true, alt: "the fraction 2 over 3 the square root of 72 plus the fraction 1 over 5 the square root of 50")[$frac(2, 3) sqrt(72) + frac(1, 5) sqrt(50)$]

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

#math.equation(block: true, alt: "the fraction 2 over 5 the square root of 75 plus the fraction 3 over 4 the square root of 48")[$frac(2, 5) sqrt(75) + frac(3, 4) sqrt(48)$]

#math.equation(block: true, alt: "the fraction 1 over 2 the square root of 20 minus the fraction 2 over 3 the square root of 45")[$frac(1, 2) sqrt(20) − frac(2, 3) sqrt(45)$]

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

#math.equation(block: true, alt: "the fraction 2 over 3 the square root of 54 minus the fraction 3 over 4 the square root of 96")[$frac(2, 3) sqrt(54) − frac(3, 4) sqrt(96)$]

#math.equation(block: true, alt: "the fraction 1 over 6 the square root of 27 minus the fraction 3 over 8 the square root of 48")[$frac(1, 6) sqrt(27) − frac(3, 8) sqrt(48)$]

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

#math.equation(block: true, alt: "the fraction 1 over 8 the square root of 32 minus the fraction 1 over 10 the square root of 50")[$frac(1, 8) sqrt(32) − frac(1, 10) sqrt(50)$]

#math.equation(block: true, alt: "the fraction 1 over 4 the square root of 98 minus the fraction 1 over 3 the square root of 128")[$frac(1, 4) sqrt(98) − frac(1, 3) sqrt(128)$]

#solutionbox[
#math.equation(block: true, alt: "minus the fraction 11 over 12 the square root of 2")[$− frac(11, 12) sqrt(2)$]
]

#math.equation(block: true, alt: "the fraction 1 over 3 the square root of 24 plus the fraction 1 over 4 the square root of 54")[$frac(1, 3) sqrt(24) + frac(1, 4) sqrt(54)$]

#math.equation(block: true, alt: "the square root of 72 a to the power 5 minus the square root of 50 a to the power 5")[$sqrt(72 a^(5)) − sqrt(50 a^(5))$]

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

#math.equation(block: true, alt: "the square root of 48 b to the power 5 minus the square root of 75 b to the power 5")[$sqrt(48 b^(5)) − sqrt(75 b^(5))$]

#math.equation(block: true, alt: "the square root of 80 c to the power 7 minus the square root of 20 c to the power 7")[$sqrt(80 c^(7)) − sqrt(20 c^(7))$]

#solutionbox[
#math.equation(block: true, alt: "2 c cubed the square root of 5 c")[$2 c^(3) sqrt(5 c)$]
]

#math.equation(block: true, alt: "the square root of 96 d to the power 9 minus the square root of 24 d to the power 9")[$sqrt(96 d^(9)) − sqrt(24 d^(9))$]

#math.equation(block: true, alt: "9 the square root of 80 p to the power 4 minus 6 the square root of 98 p to the power 4")[$9 sqrt(80 p^(4)) − 6 sqrt(98 p^(4))$]

#solutionbox[
#math.equation(block: true, alt: "36 p squared the square root of 5 minus 42 p squared the square root of 2")[$36 p^(2) sqrt(5) − 42 p^(2) sqrt(2)$]
]

#math.equation(block: true, alt: "8 the square root of 72 q to the power 6 minus 3 the square root of 75 q to the power 6")[$8 sqrt(72 q^(6)) − 3 sqrt(75 q^(6))$]

#math.equation(block: true, alt: "2 the square root of 50 r to the power 8 plus 4 the square root of 54 r to the power 8")[$2 sqrt(50 r^(8)) + 4 sqrt(54 r^(8))$]

#solutionbox[
#math.equation(block: true, alt: "10 r to the power 4 the square root of 2 plus 12 r to the power 4 the square root of 6")[$10 r^(4) sqrt(2) + 12 r^(4) sqrt(6)$]
]

#math.equation(block: true, alt: "5 the square root of 27 s to the power 6 plus 2 the square root of 20 s to the power 6")[$5 sqrt(27 s^(6)) + 2 sqrt(20 s^(6))$]

#math.equation(block: true, alt: "3 the square root of 20 x squared minus 4 the square root of 45 x squared plus 5 x the square root of 80")[$3 sqrt(20 x^(2)) − 4 sqrt(45 x^(2)) + 5 x sqrt(80)$]

#solutionbox[
#math.equation(block: true, alt: "14 x the square root of 5")[$14 x sqrt(5)$]
]

#math.equation(block: true, alt: "2 the square root of 28 x squared minus the square root of 63 x squared plus 6 x the square root of 7")[$2 sqrt(28 x^(2)) − sqrt(63 x^(2)) + 6 x sqrt(7)$]

#math.equation(block: true, alt: "3 the square root of 128 y squared plus 4 y the square root of 162 minus 8 the square root of 98 y squared")[$3 sqrt(128 y^(2)) + 4 y sqrt(162) − 8 sqrt(98 y^(2))$]

#solutionbox[
#math.equation(block: true, alt: "4 y the square root of 2")[$4 y sqrt(2)$]
]

#math.equation(block: true, alt: "3 the square root of 75 y squared plus 8 y the square root of 48 minus the square root of 300 y squared")[$3 sqrt(75 y^(2)) + 8 y sqrt(48) − sqrt(300 y^(2))$]

#strong[Mixed Practice]

#math.equation(block: true, alt: "2 the square root of 8 plus 6 the square root of 8 minus 5 the square root of 8")[$2 sqrt(8) + 6 sqrt(8) − 5 sqrt(8)$]

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

#math.equation(block: true, alt: "the fraction 2 over 3 the square root of 27 plus the fraction 3 over 4 the square root of 48")[$frac(2, 3) sqrt(27) + frac(3, 4) sqrt(48)$]

#math.equation(block: true, alt: "the square root of 175 k to the power 4 minus the square root of 63 k to the power 4")[$sqrt(175 k^(4)) − sqrt(63 k^(4))$]

#solutionbox[
#math.equation(block: true, alt: "2 k squared the square root of 7")[$2 k^(2) sqrt(7)$]
]

#math.equation(block: true, alt: "the fraction 5 over 6 the square root of 162 plus the fraction 3 over 16 the square root of 128")[$frac(5, 6) sqrt(162) + frac(3, 16) sqrt(128)$]

#math.equation(block: true, alt: "2 the square root of 363 minus 2 the square root of 300")[$2 sqrt(363) − 2 sqrt(300)$]

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

#math.equation(block: true, alt: "the square root of 150 plus 4 the square root of 6")[$sqrt(150) + 4 sqrt(6)$]

#math.equation(block: true, alt: "9 the square root of 2 minus 8 the square root of 2")[$9 sqrt(2) − 8 sqrt(2)$]

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

#math.equation(block: true, alt: "5 the square root of x minus 8 the square root of y")[$5 sqrt(x) − 8 sqrt(y)$]

#math.equation(block: true, alt: "8 the square root of 13 minus 4 the square root of 13 minus 3 the square root of 13")[$8 sqrt(13) − 4 sqrt(13) − 3 sqrt(13)$]

#solutionbox[
#math.equation(block: true, alt: "the square root of 13")[$sqrt(13)$]
]

#math.equation(block: true, alt: "5 the square root of 12 c to the power 4 minus 3 the square root of 27 c to the power 6")[$5 sqrt(12 c^(4)) − 3 sqrt(27 c^(6))$]

#math.equation(block: true, alt: "the square root of 80 a to the power 5 minus the square root of 45 a to the power 5")[$sqrt(80 a^(5)) − sqrt(45 a^(5))$]

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

#math.equation(block: true, alt: "the fraction 3 over 5 the square root of 75 minus the fraction 1 over 4 the square root of 48")[$frac(3, 5) sqrt(75) − frac(1, 4) sqrt(48)$]

#math.equation(block: true, alt: "21 the square root of 19 minus 2 the square root of 19")[$21 sqrt(19) − 2 sqrt(19)$]

#solutionbox[
#math.equation(block: true, alt: "19 the square root of 19")[$19 sqrt(19)$]
]

#math.equation(block: true, alt: "the square root of 500 plus the square root of 405")[$sqrt(500) + sqrt(405)$]

#math.equation(block: true, alt: "the fraction 5 over 6 the square root of 27 plus the fraction 5 over 8 the square root of 48")[$frac(5, 6) sqrt(27) + frac(5, 8) sqrt(48)$]

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

#math.equation(block: true, alt: "11 the square root of 11 minus 10 the square root of 11")[$11 sqrt(11) − 10 sqrt(11)$]

#math.equation(block: true, alt: "the square root of 75 minus the square root of 108")[$sqrt(75) − sqrt(108)$]

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

#math.equation(block: true, alt: "2 the square root of 98 minus 4 the square root of 72")[$2 sqrt(98) − 4 sqrt(72)$]

#math.equation(block: true, alt: "4 the square root of 24 x squared minus the square root of 54 x squared plus 3 x the square root of 6")[$4 sqrt(24 x^(2)) − sqrt(54 x^(2)) + 3 x sqrt(6)$]

#solutionbox[
#math.equation(block: true, alt: "8 x the square root of 6")[$8 x sqrt(6)$]
]

#math.equation(block: true, alt: "8 the square root of 80 y to the power 6 minus 6 the square root of 48 y to the power 6")[$8 sqrt(80 y^(6)) − 6 sqrt(48 y^(6))$]


==== Everyday Math

A decorator decides to use square tiles as an accent strip in the design of a new shower, but she wants to rotate the tiles to look like diamonds. She will use 9 large tiles that measure 8 inches on a side and 8 small tiles that measure 2 inches on a side. Determine the width of the accent strip by simplifying the expression #math.equation(block: false, alt: "9 open parenthesis 8 the square root of 2 close parenthesis plus 8 open parenthesis 2 the square root of 2 close parenthesis")[$9 ( 8 sqrt(2) ) + 8 ( 2 sqrt(2) )$]. (Round to the nearest tenth of an inch.)

#solutionbox[
#math.equation(block: true, alt: "124.5 inches")[$124.5 #h(0.2em) "inches"$]
]

Suzy wants to use square tiles on the border of a spa she is installing in her backyard. She will use large tiles that have area of 12 square inches, medium tiles that have area of 8 square inches, and small tiles that have area of 4 square inches. Once section of the border will require 4 large tiles, 8 medium tiles, and 10 small tiles to cover the width of the wall. Simplify the expression #math.equation(block: false, alt: "4 the square root of 12 plus 8 the square root of 8 plus 10 the square root of 4")[$4 sqrt(12) + 8 sqrt(8) + 10 sqrt(4)$] to determine the width of the wall. (Round to the nearest tenth of an inch.)


==== Writing Exercises

Explain the difference between like radicals and unlike radicals. Make sure your answer makes sense for radicals containing both numbers and variables.

#solutionbox[
Answers will vary.
]

Explain the process for determining whether two radicals are like or unlike. Make sure your answer makes sense for radicals containing both numbers and variables.


==== 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 – I don’t get it!” Under the “I can…” column the rows read, “add and subtract like square roots.,” and “add and subtract square roots that need simplification.” 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 – I don’t get it!” Under the “I can…” column the rows read, “add and subtract like square roots.,” and “add and subtract square roots that need simplification.” The other rows under the other columns are empty.", caption: none)

ⓑ What does this checklist tell you about your mastery of this section? What steps will you take to improve?