9.4 Multiply Square Roots

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

$\sqrt{a b} = \sqrt{a} \cdot \sqrt{b}$

We can use the Product Property of Square Roots ‘in reverse’ to multiply square roots.

$\sqrt{a} \cdot \sqrt{b} = \sqrt{a b}$

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.

So we can multiply $\sqrt{3} \cdot \sqrt{5}$ in this way:

$\begin{matrix} \sqrt{3} \cdot \sqrt{5} \\ \sqrt{3 \cdot 5} \\ \sqrt{15} \end{matrix}$

Sometimes the product gives us a perfect square:

$\begin{matrix} \sqrt{2} \cdot \sqrt{8} \\ \sqrt{2 \cdot 8} \\ \sqrt{16} \\ 4 \end{matrix}$

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 $4 x times 3 y$ we multiply the coefficients together and then the variables. The result is $12 x y$ . Keep this in mind as you do these examples.

Example 1

Simplify: ⓐ $\sqrt{2} \cdot \sqrt{6}$ ⓑ $(4 \sqrt{3}) (2 \sqrt{12})$ .

ⓐ

	$\sqrt{2} \cdot \sqrt{6}$
Multiply using the Product Property.	$the square root of 12$
Simplify the radical.	$\sqrt{4} \cdot \sqrt{3}$
Simplify.	$2 the square root of 3$

ⓑ

	$(4 \sqrt{3}) (2 \sqrt{12})$
Multiply using the Product Property.	$8 the square root of 36$
Simplify the radical.	$8 times 6$
Simplify.	$48$

Notice that in (b) we multiplied the coefficients and multiplied the radicals. Also, we did not simplify $the square root of 12$ . We waited to get the product and then simplified.

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

Example 3

Simplify: ⓐ $(\sqrt{8 x^{3}}) (\sqrt{3 x})$ ⓑ $(\sqrt{20 y^{2}}) (\sqrt{5 y^{3}})$ .

ⓐ

	$(\sqrt{8 x^{3}}) (\sqrt{3 x})$
Multiply using the Product Property.	$\sqrt{24 x^{4}}$
Simplify the radical.	$\sqrt{4 x^{4}} \cdot \sqrt{6}$
Simplify.	$2 x squared the square root of 6$

ⓑ

	$(\sqrt{20 y^{2}}) (\sqrt{5 y^{3}})$
Multiply using the Product Property.	$\sqrt{100 y^{5}}$
Simplify the radical.	$10 y squared the square root of y$

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

By realizing that squaring and taking a square root are ‘opposite’ operations, we can simplify $open parenthesis the square root of 2 close parenthesis squared$ and get 2 right away. When we multiply the two like square roots in part (a) of the next example, it is the same as squaring.

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.

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.

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.

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

Example 13

Simplify: ⓐ ${(2 + \sqrt{3})}^{2}$ ⓑ ${(4 - 2 \sqrt{5})}^{2}$ .

Be sure to include the $2 a b$ term when squaring a binomial.

ⓐ

Multiply using the binomial square pattern.
Simplify.
Combine like terms.
ⓑ

Multiply using the binomial square pattern.
Simplify.
Combine like terms.

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

Key Concepts

Product Property of Square Roots If a, b are nonnegative real numbers, then

$\sqrt{a b} = \sqrt{a} \cdot \sqrt{b} and \sqrt{a} \cdot \sqrt{b} = \sqrt{a b}$
Special formulas for multiplying binomials and conjugates:

$\begin{matrix} {(a + b)}^{2} = a^{2} + 2 a b + b^{2} & (a - b) (a + b) = a^{2} - b^{2} \\ {(a - b)}^{2} = a^{2} - 2 a b + b^{2} \end{matrix}$
The FOIL method can be used to multiply binomials containing radicals.

Practice Makes Perfect

Multiply Square Roots

In the following exercises, simplify.

ⓐ $\sqrt{2} \cdot \sqrt{8}$ ⓑ $(3 \sqrt{3}) (2 \sqrt{18})$

ⓐ $4$ ⓑ $18 the square root of 6$

ⓐ $\sqrt{6} \cdot \sqrt{6}$ ⓑ $(3 \sqrt{2}) (2 \sqrt{32})$

ⓐ $\sqrt{7} \cdot \sqrt{14}$ ⓑ $(4 \sqrt{8}) (5 \sqrt{8})$

ⓐ $7 the square root of 2$ ⓑ 160

ⓐ $\sqrt{6} \cdot \sqrt{12}$ ⓑ $(2 \sqrt{5}) (2 \sqrt{10})$

$(5 \sqrt{2}) (3 \sqrt{6})$

$30 the square root of 3$

$(2 \sqrt{3}) (4 \sqrt{6})$

$(−2 \sqrt{3}) (3 \sqrt{18})$

$−18 the square root of 6$

$(−4 \sqrt{5}) (5 \sqrt{10})$

$(5 \sqrt{6}) (− \sqrt{12})$

$−30 the square root of 2$

$(6 \sqrt{2}) (− \sqrt{10})$

$(−2 \sqrt{7}) (−2 \sqrt{14})$

$28 the square root of 2$

$(−2 \sqrt{11}) (−4 \sqrt{22})$

ⓐ $(\sqrt{15 y}) (\sqrt{5 y^{3}})$ ⓑ $(\sqrt{2 n^{2}}) (\sqrt{18 n^{3}})$

ⓐ $5 y squared the square root of 3$ ⓑ $6 n squared the square root of n$

ⓐ $(\sqrt{14 x^{3}}) (\sqrt{7 x^{3}})$ ⓑ $(\sqrt{3 q^{2}}) (\sqrt{48 q^{3}})$

ⓐ $(\sqrt{16 y^{2}}) (\sqrt{8 y^{4}})$ ⓑ $(\sqrt{11 s^{6}}) (\sqrt{11 s})$

ⓐ $8 y cubed the square root of 2$ ⓑ $11 s cubed the square root of s$

ⓐ $(\sqrt{8 x^{3}}) (\sqrt{3 x})$ ⓑ $(\sqrt{7 r}) (\sqrt{7 r^{8}})$

$(2 \sqrt{5 b^{3}}) (4 \sqrt{15 b})$

$40 b squared the square root of 3$

$(3 \sqrt{8 c^{5}}) (2 \sqrt{6 c^{3}})$

$(5 \sqrt{2 d^{7}}) (3 \sqrt{50 d^{3}})$

$150 d to the power 5$

$(4 \sqrt{6 t^{2}}) (3 \sqrt{3 t^{2}})$

$(3 \sqrt{4 y^{4}}) (3 \sqrt{9 y^{5}})$

$54 y^{4} \sqrt{y}$

$(−2 \sqrt{7 z^{3}}) (3 \sqrt{14 z^{8}})$

$(4 \sqrt{2 k^{5}}) (−3 \sqrt{32 k^{6}})$

$−96 k^{5} \sqrt{k}$

ⓐ $open parenthesis the square root of 7 close parenthesis squared$ ⓑ $open parenthesis − the square root of 15 close parenthesis squared$

ⓐ $open parenthesis the square root of 11 close parenthesis squared$ ⓑ $open parenthesis − the square root of 21 close parenthesis squared$

ⓐ 11 ⓑ 21

ⓐ $open parenthesis the square root of 19 close parenthesis squared$ ⓑ $open parenthesis − the square root of 5 close parenthesis squared$

ⓐ $open parenthesis the square root of 23 close parenthesis squared$
ⓑ $open parenthesis − the square root of 3 close parenthesis squared$

ⓐ 23 ⓑ 3

ⓐ $(4 \sqrt{11}) (−3 \sqrt{11})$ ⓑ $open parenthesis 5 the square root of 3 close parenthesis squared$

ⓐ $(2 \sqrt{13}) (−9 \sqrt{13})$ ⓑ $open parenthesis 6 the square root of 5 close parenthesis squared$

ⓐ −234 ⓑ 180

ⓐ $(−3 \sqrt{12}) (−2 \sqrt{6})$ ⓑ $open parenthesis −4 the square root of 10 close parenthesis squared$

ⓐ $(−7 \sqrt{5}) (−3 \sqrt{10})$ ⓑ $open parenthesis −2 the square root of 14 close parenthesis squared$

ⓐ $105 the square root of 2$ ⓑ 56

Use Polynomial Multiplication to Multiply Square Roots

In the following exercises, simplify.

ⓐ $3 (4 - \sqrt{3})$ ⓑ $\sqrt{2} (4 - \sqrt{6})$

ⓐ $4 (6 - \sqrt{11})$ ⓑ $\sqrt{2} (5 - \sqrt{12})$

ⓐ $24 - 4 \sqrt{11}$ ⓑ $5 \sqrt{2} - 2 \sqrt{6}$

ⓐ $5 (3 - \sqrt{7})$ ⓑ $\sqrt{3} (4 - \sqrt{15})$

ⓐ $7 (−2 - \sqrt{11})$ ⓑ $\sqrt{7} (6 - \sqrt{14})$

ⓐ $−14 - 7 \sqrt{11}$ ⓑ $6 \sqrt{7} - 7 \sqrt{2}$

ⓐ $\sqrt{7} (5 + 2 \sqrt{7})$ ⓑ $\sqrt{5} (\sqrt{10} + \sqrt{18})$

ⓐ $\sqrt{11} (8 + 4 \sqrt{11})$ ⓑ $\sqrt{3} (\sqrt{12} + \sqrt{27})$

ⓐ $44 + 8 \sqrt{11}$ ⓑ $15$

ⓐ $\sqrt{11} (−3 + 4 \sqrt{11})$ ⓑ $\sqrt{3} (\sqrt{15} - \sqrt{18})$

ⓐ $\sqrt{2} (−5 + 9 \sqrt{2})$ ⓑ $\sqrt{7} (\sqrt{3} - \sqrt{21})$

ⓐ $18 - 5 \sqrt{2}$ ⓑ $\sqrt{21} - 7 \sqrt{3}$

$(8 + \sqrt{3}) (2 - \sqrt{3})$

$(7 + \sqrt{3}) (9 - \sqrt{3})$

$60 + 2 \sqrt{3}$

$(8 - \sqrt{2}) (3 + \sqrt{2})$

$(9 - \sqrt{2}) (6 + \sqrt{2})$

$52 + 3 \sqrt{2}$

$(3 - \sqrt{7}) (5 - \sqrt{7})$

$(5 - \sqrt{7}) (4 - \sqrt{7})$

$27 - 9 \sqrt{7}$

$(1 + 3 \sqrt{10}) (5 - 2 \sqrt{10})$

$(7 - 2 \sqrt{5}) (4 + 9 \sqrt{5})$

$−62 + 55 \sqrt{5}$

$(\sqrt{3} + \sqrt{10}) (\sqrt{3} + 2 \sqrt{10})$

$(\sqrt{11} + \sqrt{5}) (\sqrt{11} + 6 \sqrt{5})$

$41 + 7 \sqrt{55}$

$(2 \sqrt{7} - 5 \sqrt{11}) (4 \sqrt{7} + 9 \sqrt{11})$

$(4 \sqrt{6} + 7 \sqrt{13}) (8 \sqrt{6} - 3 \sqrt{13})$

$−81 + 44 \sqrt{78}$

$(5 - \sqrt{u}) (3 + \sqrt{u})$

$(9 - \sqrt{w}) (2 + \sqrt{w})$

$18 + 7 \sqrt{w} - w$

$(7 + 2 \sqrt{m}) (4 + 9 \sqrt{m})$

$(6 + 5 \sqrt{n}) (11 + 3 \sqrt{n})$

$66 + 73 \sqrt{n} + 15 n$

ⓐ ${(3 + \sqrt{5})}^{2}$
ⓑ ${(2 - 5 \sqrt{3})}^{2}$

ⓐ ${(4 + \sqrt{11})}^{2}$ ⓑ ${(3 - 2 \sqrt{5})}^{2}$

ⓐ $27 + 8 \sqrt{11}$ ⓑ $29 - 12 \sqrt{5}$

ⓐ ${(9 - \sqrt{6})}^{2}$ ⓑ ${(10 + 3 \sqrt{7})}^{2}$

ⓐ ${(5 - \sqrt{10})}^{2}$ ⓑ ${(8 + 3 \sqrt{2})}^{2}$

ⓐ $35 - 10 \sqrt{10}$ ⓑ $82 + 48 \sqrt{2}$

$(3 - \sqrt{5}) (3 + \sqrt{5})$

$(10 - \sqrt{3}) (10 + \sqrt{3})$

$(4 + \sqrt{2}) (4 - \sqrt{2})$

$(7 + \sqrt{10}) (7 - \sqrt{10})$

$(4 + 9 \sqrt{3}) (4 - 9 \sqrt{3})$

$(1 + 8 \sqrt{2}) (1 - 8 \sqrt{2})$

$−127$

$(12 - 5 \sqrt{5}) (12 + 5 \sqrt{5})$

$(9 - 4 \sqrt{3}) (9 + 4 \sqrt{3})$

Mixed Practice

In the following exercises, simplify.

$\sqrt{3} \cdot \sqrt{21}$

$(4 \sqrt{6}) (− \sqrt{18})$

$−24 the square root of 3$

$(−5 + \sqrt{7}) (6 + \sqrt{21})$

$(−5 \sqrt{7}) (6 \sqrt{21})$

$−210 the square root of 3$

$(−4 \sqrt{2}) (2 \sqrt{18})$

$(\sqrt{35 y^{3}}) (\sqrt{7 y^{3}})$

$7 y cubed the square root of 5$

$(4 \sqrt{12 x^{5}}) (2 \sqrt{6 x^{3}})$

$open parenthesis the square root of 29 close parenthesis squared$

$29$

$(−4 \sqrt{17}) (−3 \sqrt{17})$

$(−4 + \sqrt{17}) (−3 + \sqrt{17})$

$29 - 7 \sqrt{17}$

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.

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.

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.
ⓑ Find the exact area of the face of the square base.

ⓐ $4.6 feet$ ⓑ $21 sq. feet$

A square garden will be made with a stone border on one edge. If only $3 + \sqrt{10}$ feet of stone are available, simplify ${(3 + \sqrt{10})}^{2}$ 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 $\sqrt{5} + \sqrt{6}$ yards and one section with side length $\sqrt{2} + \sqrt{3}$ yards. Simplify ${(\sqrt{5} + \sqrt{6})}^{2} + {(\sqrt{2} + \sqrt{3})}^{2}$ to determine the total area of the garden. Round your answer to the nearest tenth.

$31.9 square yards$

Suppose a third section will be added to the garden in the previous exercise. The third section is to have a width of $the square root of 432$ yards. Write an expression that gives the total area of the garden.

Writing Exercises

ⓐ Explain why $open parenthesis − the square root of n close parenthesis squared$ is always positive, for $n greater than or equal to 0$ .
ⓑ Explain why $- {(\sqrt{n})}^{2}$ is always negative, for $n greater than or equal to 0$ .

ⓐ 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 ${(3 + \sqrt{2})}^{2}$ . Explain all your steps.

Self Check

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

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.

ⓑ 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?

	$(6 \sqrt{2}) (3 \sqrt{10})$
Multiply using the Product Property.	$18 the square root of 20$
Simplify the radical.	$18 \sqrt{4} \cdot \sqrt{5}$
Simplify.	$18 \cdot 2 \cdot \sqrt{5}$
	$36 the square root of 5$

	$(10 \sqrt{6 p^{3}}) (3 \sqrt{18 p})$
Multiply.	$30 \sqrt{108 p^{4}}$
Simplify the radical.	$30 \sqrt{36 p^{4}} \cdot \sqrt{3}$
	$30 \cdot 6 p^{2} \cdot \sqrt{3}$
	$180 p squared the square root of 3$

	$open parenthesis the square root of 2 close parenthesis squared$
Rewrite as a product.	$(\sqrt{2}) (\sqrt{2})$
Multiply.	$the square root of 4$
Simplify.	$2$

	$open parenthesis − the square root of 11 close parenthesis squared$
Rewrite as a product.	$(− \sqrt{11}) (− \sqrt{11})$
Multiply.	$the square root of 121$
Simplify.	$11$

	$(2 \sqrt{3}) (8 \sqrt{3})$
Multiply. Remember, ${(\sqrt{3})}^{2} = 3 .$	$16 times 3$
Simplify.	$48$

	$\sqrt{2} (4 - \sqrt{10})$
Distribute.	$4 \sqrt{2} - \sqrt{20}$
	$4 \sqrt{2} - 2 \sqrt{5}$

	$\sqrt{5} (7 + 2 \sqrt{5})$
Multiply.	$7 \sqrt{5} + 2 \cdot 5$
Simplify.	$7 \sqrt{5} + 10$
	$10 + 7 \sqrt{5}$

	$\sqrt{6} (\sqrt{2} + \sqrt{18})$
Multiply.	$\sqrt{12} + \sqrt{108}$
Simplify.	$\sqrt{4} \cdot \sqrt{3} + \sqrt{36} \cdot \sqrt{3}$
	$2 \sqrt{3} + 6 \sqrt{3}$
Combine like radicals.	$8 the square root of 3$

	$(2 + \sqrt{3}) (4 - \sqrt{3})$
Multiply.	$8 - 2 \sqrt{3} + 4 \sqrt{3} - 3$
Combine like terms.	$5 + 2 \sqrt{3}$

	$(3 - 2 \sqrt{7}) (4 - 2 \sqrt{7})$
Multiply.	$12 - 6 \sqrt{7} - 8 \sqrt{7} + 4 \cdot 7$
Simplify.	$12 - 6 \sqrt{7} - 8 \sqrt{7} + 28$
Combine like terms.	$40 - 14 \sqrt{7}$

	$(3 \sqrt{2} - \sqrt{5}) (\sqrt{2} + 4 \sqrt{5})$
Multiply.	$3 \cdot 2 + 12 \sqrt{10} - \sqrt{10} - 4 \cdot 5$
Simplify.	$6 + 12 \sqrt{10} - \sqrt{10} - 20$
Combine like terms.	$−14 + 11 \sqrt{10}$

	$(4 - 2 \sqrt{x}) (1 + 3 \sqrt{x})$
Multiply.	$4 + 12 \sqrt{x} - 2 \sqrt{x} - 6 x$
Combine like terms.	$4 + 10 \sqrt{x} - 6 x$


Multiply using the binomial square pattern.
Simplify.
Combine like terms.


Multiply using the binomial square pattern.
Simplify.	$A fraction displaying the operation (25 - 4 * 3) divided by 13. Following the order of operations, the numerator simplifies to 25 - 12 = 13, making the fraction 13/13, which equals 1.$