Multiplication of radical expressions

Algebra ii for the community Page 1 / 1

This module is from Elementary Algebra by Denny Burzynski and Wade Ellis, Jr. The distinction between the principal square root of the number x and the secondary square root of the number x is made by explanation and by example. The simplification of the radical expressions that both involve and do not involve fractions is shown in many detailed examples; this is followed by an explanation of how and why radicals are eliminated from the denominator of a radical expression. Real-life applications of radical equations have been included, such as problems involving daily output, daily sales, electronic resonance frequency, and kinetic energy.Objectives of this module: be able to use the product property of square roots to multiply square roots.

Overview

The Product Property of Square Roots
Multiplication Rule for Square Root Expressions

The product property of square roots

In our work with simplifying square root expressions, we noted that

$\sqrt{x y} = \sqrt{x} \sqrt{y}$

Since this is an equation, we may write it as

$\sqrt{x} \sqrt{y} = \sqrt{x y}$

To multiply two square root expressions, we use the product property of square roots.

The product property $\sqrt{x} \sqrt{y} = \sqrt{x y}$

\sqrt{x} \sqrt{y} = \sqrt{x y}

The product of the square roots is the square root of the product.

In practice, it is usually easier to simplify the square root expressions before actually performing the multiplication. To see this, consider the following product:

$\sqrt{8} \sqrt{48}$
We can multiply these square roots in either of two ways:

Simplify then multiply.

$\sqrt{4 \cdot 2} \sqrt{16 \cdot 3} = (2 \sqrt{2}) (4 \sqrt{3}) = 2 \cdot 4 \sqrt{2 \cdot 3} = 8 \sqrt{6}$

Multiply then simplify.

$\sqrt{8} \sqrt{48} = \sqrt{8 \cdot 48} = \sqrt{384} = \sqrt{64 \cdot 6} = 8 \sqrt{6}$

Notice that in the second method, the expanded term (the third expression, $\sqrt{384}$ ) may be difficult to factor into a perfect square and some other number.

Multiplication rule for square root expressions

The preceding example suggests that the following rule for multiplying two square root expressions.

Rule for multiplying square root expressions

Simplify each square root expression, if necessary.
Perform the multiplecation.
Simplify, if necessary.

Sample set a

Find each of the following products.

$\sqrt{3} \sqrt{6} = \sqrt{3 \cdot 6} = \sqrt{18} = \sqrt{9 \cdot 2} = 3 \sqrt{2}$

$\sqrt{8} \sqrt{2} = 2 \sqrt{2} \sqrt{2} = 2 \sqrt{2 \cdot 2} = 2 \sqrt{4} = 2 \cdot 2 = 4$

This product might be easier if we were to multiply first and then simplify.

$\sqrt{8} \sqrt{2} = \sqrt{8 \cdot 2} = \sqrt{16} = 4$

$\sqrt{20} \sqrt{7} = \sqrt{4} \sqrt{5} \sqrt{7} = 2 \sqrt{5 \cdot 7} = 2 \sqrt{35}$

$\begin{array}{l} \sqrt{5 a^{3}} \sqrt{27 a^{5}} = (a \sqrt{5 a}) (3 a^{2} \sqrt{3 a}) & = & 3 a^{3} \sqrt{15 a^{2}} \\ = & 3 a^{3} \cdot a \sqrt{15} \\ = & 3 a^{4} \sqrt{15} \end{array}$

$\begin{array}{l} \sqrt{{(x + 2)}^{7}} \sqrt{x - 1} = \sqrt{{(x + 2)}^{6} (x + 2)} \sqrt{x - 1} & = & {(x + 2)}^{3} \sqrt{(x + 2)} \sqrt{x - 1} \\ = & {(x + 2)}^{3} \sqrt{(x + 2) (x - 1)} \\ \begin{matrix} \begin{matrix} \begin{matrix} or \end{matrix} \end{matrix} \end{matrix} & = & {(x + 2)}^{3} \sqrt{x^{2} + x - 2} \end{array}$

Finding the product of the square root of three and the binomial seven plus the square root of six, using the rule for multiplying square root expressions. See the longdesc for a full description.

Finding the product of the square root of six and the binomial the square root of two minus the square root of ten, using the rule for multiplying square root expressions. See the longdesc for a full description.

Finding the product of two expressions involving square roots. The first expression is the square root of forty-five a to the sixth power b cubed. The second expression is a binomial with the square root of 'five times the cube of the binomial b minus three. See the longdesc for a full description.

Practice set a

Find each of the following products.

$\sqrt{5} \sqrt{6}$

$\sqrt{30}$

$\sqrt{32} \sqrt{2}$

$\sqrt{x + 4} \sqrt{x + 3}$

$\sqrt{(x + 4) (x + 3)}$

$\sqrt{8 m^{5} n} \sqrt{20 m^{2} n}$

$4 m^{3} n \sqrt{10 m}$

$\sqrt{9 {(k - 6)}^{3}} \sqrt{k^{2} - 12 k + 36}$

$3 {(k - 6)}^{2} \sqrt{k - 6}$

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

$\sqrt{6} + \sqrt{15}$

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

$a \sqrt{10} - 4 a^{2}$

$\sqrt{32 m^{5} n^{8}} (\sqrt{2 m n^{2}} - \sqrt{10 n^{7}})$

$8 m^{3} n^{2} \sqrt{n} - 8 m^{2} n^{5} \sqrt{5 m}$

Exercises

$\sqrt{2} \sqrt{10}$

$2 \sqrt{5}$

$\sqrt{3} \sqrt{15}$

$\sqrt{7} \sqrt{8}$

$2 \sqrt{14}$

$\sqrt{20} \sqrt{3}$

$\sqrt{32} \sqrt{27}$

$12 \sqrt{6}$

$\sqrt{45} \sqrt{50}$

$\sqrt{5} \sqrt{5}$

$\sqrt{7} \sqrt{7}$

$\sqrt{8} \sqrt{8}$

$\sqrt{15} \sqrt{15}$

$\sqrt{48} \sqrt{27}$

$\sqrt{80} \sqrt{20}$

$\sqrt{5} \sqrt{m}$

$\sqrt{5 m}$

$\sqrt{7} \sqrt{a}$

$\sqrt{6} \sqrt{m}$

$\sqrt{6 m}$

$\sqrt{10} \sqrt{h}$

$\sqrt{20} \sqrt{a}$

$2 \sqrt{5 a}$

$\sqrt{48} \sqrt{x}$

$\sqrt{75} \sqrt{y}$

$5 \sqrt{3 y}$

$\sqrt{200} \sqrt{m}$

$\sqrt{a} \sqrt{a}$

$a$

$\sqrt{x} \sqrt{x}$

$\sqrt{y} \sqrt{y}$

$y$

$\sqrt{h} \sqrt{h}$

$\sqrt{3} \sqrt{3}$

$\sqrt{6} \sqrt{6}$

$\sqrt{k} \sqrt{k}$

$k$

$\sqrt{m} \sqrt{m}$

$\sqrt{m^{2}} \sqrt{m}$

$m \sqrt{m}$

$\sqrt{a^{2}} \sqrt{a}$

$\sqrt{x^{3}} \sqrt{x}$

$x^{2}$

$\sqrt{y^{3}} \sqrt{y}$

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

$y^{2} \sqrt{y}$

$\sqrt{k} \sqrt{k^{6}}$

$\sqrt{a^{3}} \sqrt{a^{5}}$

$a^{4}$

$\sqrt{x^{3}} \sqrt{x^{7}}$

$\sqrt{x^{9}} \sqrt{x^{3}}$

$x^{6}$

$\sqrt{y^{7}} \sqrt{y^{9}}$

$\sqrt{y^{3}} \sqrt{y^{4}}$

$y^{3} \sqrt{y}$

$\sqrt{x^{8}} \sqrt{x^{5}}$

$\sqrt{x + 2} \sqrt{x - 3}$

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

$\sqrt{a - 6} \sqrt{a + 1}$

$\sqrt{y + 3} \sqrt{y - 2}$

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

$\sqrt{h + 1} \sqrt{h - 1}$

$\sqrt{x + 9} \sqrt{{(x + 9)}^{2}}$

$(x + 9) \sqrt{x + 9}$

$\sqrt{y - 3} \sqrt{{(y - 3)}^{5}}$

$\sqrt{3 a^{2}} \sqrt{15 a^{3}}$

$3 a^{2} \sqrt{5 a}$

$\sqrt{2 m^{4} n^{3}} \sqrt{14 m^{5} n}$

$\sqrt{12 {(p - q)}^{3}} \sqrt{3 {(p - q)}^{5}}$

$6 {(p - q)}^{4}$

$\sqrt{15 a^{2} {(b + 4)}^{4}} \sqrt{21 a^{3} {(b + 4)}^{5}}$

$\sqrt{125 m^{5} n^{4} r^{8}} \sqrt{8 m^{6} r}$

$10 m^{5} n^{2} r^{4} \sqrt{10 m r}$

$\sqrt{7 {(2 k - 1)}^{11} {(k + 1)}^{3}} \sqrt{14 {(2 k - 1)}^{10}}$

$\sqrt{y^{3}} \sqrt{y^{5}} \sqrt{y^{2}}$

$y^{5}$

$\sqrt{x^{6}} \sqrt{x^{2}} \sqrt{x^{9}}$

$\sqrt{2 a^{4}} \sqrt{5 a^{3}} \sqrt{2 a^{7}}$

$2 a^{7} \sqrt{5}$

$\sqrt{x^{n}} \sqrt{x^{n}}$

$\sqrt{y^{2} n} \sqrt{y^{4} n}$

$y^{3 n}$

$\sqrt{a^{2 n + 5}} \sqrt{a^{3}}$

$\sqrt{2 m^{3 n + 1}} \sqrt{10 m^{n + 3}}$

$2 m^{2 n + 2} \sqrt{5}$

$\sqrt{75 {(a - 2)}^{7}} \sqrt{48 a - 96}$

$\sqrt{2} (\sqrt{8} + \sqrt{6})$

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

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

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

$\sqrt{3 x} + \sqrt{6}$

$\sqrt{11} (\sqrt{y} + \sqrt{3})$

$\sqrt{8} (\sqrt{a} - \sqrt{3 a})$

$2 \sqrt{2 a} - 2 \sqrt{6 a}$

$\sqrt{x} (\sqrt{x^{3}} - \sqrt{2 x^{4}})$

$\sqrt{y} (\sqrt{y^{5}} + \sqrt{3 y^{3}})$

$y^{2} (y + \sqrt{3})$

$\sqrt{8 a^{5}} (\sqrt{2 a} - \sqrt{6 a^{11}})$

$\sqrt{12 m^{3}} (\sqrt{6 m^{7}} - \sqrt{3 m})$

$6 m^{2} (m^{3} \sqrt{2} - 1)$

$\sqrt{5 x^{4} y^{3}} (\sqrt{8 x y} - 5 \sqrt{7 x})$

Exercises for review

( [link] ) Factor $a^{4} y^{4} - 25 w^{2} .$

$(a^{2} y^{2} + 5 w) (a^{2} y^{2} - 5 w)$

( [link] ) Find the slope of the line that passes through the points $(- 5, 4)$ and $(- 3, 4) .$

( [link] ) Perform the indicated operations:

$\frac{15 x^{2} - 20 x}{6 x^{2} + x - 12} \cdot \frac{8 x + 12}{x^{2} - 2 x - 15} \div \frac{5 x^{2} + 15 x}{x^{2} - 25}$

$\frac{4 (x + 5)}{{(x + 3)}^{2}}$

( [link] ) Simplify $\sqrt{x^{4} y^{2} z^{6}}$ by removing the radical sign.

( [link] ) Simplify $\sqrt{12 x^{3} y^{5} z^{8}} .$

$2 x y^{2} z^{4} \sqrt{3 x y}$

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Source: OpenStax, Algebra ii for the community college. OpenStax CNX. Jul 03, 2014 Download for free at http://cnx.org/content/col11671/1.1

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