Key to "In the following, simplify using absolute value signs..."

High School
Algebra
Algebra I
Piecewise-Defined Functions

lunnatican4

Answered question

2021-12-10

In the following, simplify using absolute value signs as needed.

\sqrt{96 r^{3} s^{3}}

Answer & Explanation

turtletalk75

Beginner2021-12-11Added 29 answers

Given information: An expression is given as $\sqrt{96 r^{3} s^{3}}$ .
Calculations: We have been given an expression is $\sqrt{96 r^{3} s^{3}}$ .
To simplify the radical expressions by using absolute value as needed, we must know about the radical properties and the index value of root is even or odd as shown below:
We know that, ${(a b)}^{m} = a^{m} \cdot b^{m}$ and the corresponding product property of radical expression is $\sqrt[n]{a b} = \sqrt[n]{a} \cdot \sqrt[n]{b}$ .
$\Rightarrow \sqrt{96 r^{3} s^{3}}$ [Simplify the square root of $96 r^{3} s^{3}$ with radical properties]
$\Rightarrow \sqrt{16 r^{2} \cdot s^{2} \cdot 6 r s}$ [Rewrite $96 r^{3} s^{3} = 16 r^{2} \cdot s^{2} \cdot 6 r s$ ]
$\Rightarrow \sqrt{{(4 r s)}^{2}} \cdot \sqrt{6 r s}$ [By using radical property $\sqrt[n]{a b} = \sqrt[n]{a} \cdot \sqrt[n]{b}$ ]
$\Rightarrow 2 | r s | \sqrt{6 r s}$ [ $\sqrt[n]{a^{n}} = | a |$ , if the value of index is even]
Hence, the simplification of $\sqrt{96 r^{3} s^{3}} i s 2 | r s | \sqrt{6 r s}$

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