9.4 Multiply Square Roots

Learning objectives

Learning objectives

• Multiply square roots

• Use polynomial multiplication to 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

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

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:

Sometimes the product gives us a perfect square:

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

Example 9.44

Solution

ⓐ

ⓑ

Notice that in (b) we multiplied the coefficients and multiplied the radicals. Also, we did not simplify $\sqrt{12}$. We waited to get the product and then simplified.

Example 9.45

Solution

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

Example 9.46

Solution

ⓐ

ⓑ

Example 9.47

Solution

Example 9.48

Solution

ⓐ

ⓑ

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 ${(\sqrt{2})}^{2}$ 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.

Example 9.49

Solution

ⓐ

ⓑ

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.

Example 9.50

Solution

ⓐ

ⓑ

Example 9.51

Solution

ⓐ

ⓑ

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.

Example 9.52

Solution

Example 9.53

Solution

Example 9.54

Solution

Example 9.55

Solution

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 9.56

Solution

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

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

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

Example 9.57

Solution

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

Example 9.58

Solution

Example 9.59

Solution

Section 9.4 Exercises

Section 9.4 Exercises

Multiply Square Roots

In the following exercises, simplify.

Use Polynomial Multiplication to Multiply Square Roots

In the following exercises, simplify.

Mixed Practice

In the following exercises, simplify.

Everyday Math

Writing Exercises

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

ⓑ 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?

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

2. Simplify: ⓐ $\sqrt{5}\cdot\sqrt{10}$ ⓑ $(6\sqrt{3})(5\sqrt{6})$.

3. Simplify: $(3\sqrt{2})(2\sqrt{30})$.

4. Simplify: $(3\sqrt{3})(3\sqrt{6})$.

5. Simplify: ⓐ $(\sqrt{6{x}^{3}})(\sqrt{3x})$ ⓑ $(\sqrt{2{y}^{3}})(\sqrt{50{y}^{2}})$.

1. Simplify: ⓐ $(\sqrt{6{x}^{5}})(\sqrt{2x})$ ⓑ $(\sqrt{12{y}^{2}})(\sqrt{3{y}^{5}})$.

2. Simplify: $(6\sqrt{2{x}^{2}})(8\sqrt{45{x}^{4}})$.

3. Simplify: $(2\sqrt{6{y}^{4}})(12\sqrt{30y})$.

4. Simplify: ⓐ ${(\sqrt{12})}^{2}$ ⓑ ${(-\sqrt{15})}^{2}$.

5. Simplify: ⓐ ${(\sqrt{16})}^{2}$ ⓑ ${(-\sqrt{20})}^{2}$.

1. Simplify: ⓐ $(6\sqrt{11})(5\sqrt{11})$ ⓑ ${(5\sqrt{8})}^{2}$.

2. Simplify: ⓐ $(3\sqrt{7})(10\sqrt{7})$ ⓑ ${(-4\sqrt{6})}^{2}$.

3. Simplify: ⓐ $2(3-\sqrt{5})$ ⓑ $\sqrt{3}(2-\sqrt{18})$.

4. Simplify: ⓐ $6(2+\sqrt{6})$ ⓑ $\sqrt{7}(1+\sqrt{14})$.

5. Simplify: ⓐ $\sqrt{6}(1+3\sqrt{6})$ ⓑ $\sqrt{12}(\sqrt{3}+\sqrt{24})$.

1. Simplify: ⓐ $\sqrt{8}(2-5\sqrt{8})$ ⓑ $\sqrt{14}(\sqrt{2}+\sqrt{42})$.

2. Simplify: $(1+\sqrt{6})(3-\sqrt{6})$.

3. Simplify: $(4-\sqrt{10})(2+\sqrt{10})$.

4. Simplify: $(6-3\sqrt{7})(3+4\sqrt{7})$.

5. Simplify: $(2-3\sqrt{11})(4-\sqrt{11})$.

1. Simplify: $(5\sqrt{3}-\sqrt{7})(\sqrt{3}+2\sqrt{7})$.

2. Simplify: $(\sqrt{6}-3\sqrt{8})(2\sqrt{6}+\sqrt{8})$

3. Simplify: $(6-5\sqrt{m})(2+3\sqrt{m})$.

4. Simplify: $(10+3\sqrt{n})(1-5\sqrt{n})$.

5. Simplify: ⓐ ${(10+\sqrt{2})}^{2}$ ⓑ ${(1+3\sqrt{6})}^{2}$.

1. Simplify: ⓐ ${(6-\sqrt{5})}^{2}$ ⓑ ${(9-2\sqrt{10})}^{2}$.

2. Simplify: ${(2+5\sqrt{m})}^{2}$.

3. Simplify: ${(3-4\sqrt{n})}^{2}$.

4. Simplify: $(2-\sqrt{3})(2+\sqrt{3})$.

5. Simplify: $(1+\sqrt{5})(1-\sqrt{5})$.

1. Simplify: $(3-2\sqrt{5})(3+2\sqrt{5})$.

2. Simplify: $(4+5\sqrt{7})(4-5\sqrt{7})$.

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

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

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

1. $(5\sqrt{2})(3\sqrt{6})$

2. $(-2\sqrt{3})(3\sqrt{18})$

3. $(5\sqrt{6})(-\sqrt{12})$

4. $(-2\sqrt{7})(-2\sqrt{14})$

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

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

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

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

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

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

1. ⓐ ${(\sqrt{11})}^{2}$ ⓑ ${(-\sqrt{21})}^{2}$

2. ⓐ ${(\sqrt{23})}^{2}$ⓑ ${(-\sqrt{3})}^{2}$

3. ⓐ $(2\sqrt{13})(-9\sqrt{13})$ ⓑ ${(6\sqrt{5})}^{2}$

4. ⓐ $(-7\sqrt{5})(-3\sqrt{10})$ ⓑ ${(-2\sqrt{14})}^{2}$

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

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

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

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

4. $(7+\sqrt{3})(9-\sqrt{3})$

5. $(9-\sqrt{2})(6+\sqrt{2})$

1. $(5-\sqrt{7})(4-\sqrt{7})$

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

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

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

5. $(9-\sqrt{w})(2+\sqrt{w})$

1. $(6+5\sqrt{n})(11+3\sqrt{n})$

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

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

4. $(1+8\sqrt{2})(1-8\sqrt{2})$

5. $(4\sqrt{6})(-\sqrt{18})$

1. $(-5\sqrt{7})(6\sqrt{21})$

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

3. ${(\sqrt{29})}^{2}$

4. $(-4+\sqrt{17})(-3+\sqrt{17})$

5. 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.

1. ⓐ 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.

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

1. 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 ${\left(\sqrt{5}+\sqrt{6}\right)}^{2}+{\left(\sqrt{2}+\sqrt{3}\right)}^{2}$ to determine the total area of the garden. Round your answer to the nearest tenth.

2. 1. ⓐ Explain why ${(-\sqrt{n})}^{2}$ is always positive, for $n\ge{}0$.

2. ⓑ Explain why $-{\left(\sqrt{n}\right)}^{2}$ is always negative, for $n\ge{}0$.