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## Rational Exponents and Radicals 1.3 - Solution

Before multiplying the given radical expressions, we need to answer two questions.

1. Can the expressions be multiplied?
2. If so, do absolute value symbols need to be added to the answer?

The rule regarding multiplying radical expressions states that if $\sqrt[{\color{#FF0000}{n}}]{{\color{#0000FF}{a}}}$ and $\sqrt[{\color{#FF0000}{n}}]{{\color{#009600}{b}}}$ are real numbers, then $\sqrt[{\color{#FF0000}{n}}]{{\color{#0000FF}{a}}}\cdot\sqrt[{\color{#FF0000}{n}}]{{\color{#009600}{b}}}$ $=$ $\sqrt[{\color{#FF0000}{n}}]{{\color{#0000FF}{a}}{\color{#009600}{b}}}.$ $\begin{gathered} \sqrt[{\color{#FF0000}{4}}]{{\color{#0000FF}{81x^5y^4}}}\cdot \sqrt[{\color{#FF0000}{4}}]{{\color{#009600}{32x^3y}}}=\sqrt[{\color{#FF0000}{4}}]{{\color{#0000FF}{81x^5y^4}}\cdot{\color{#009600}{32x^3y}}} \end{gathered}$ Because we are assuming that both radicals are real numbers and we can see that the given expressions have the same index, we can multiply them. Now, to answer the second question, consider the rule regarding absolute value symbols. $\begin{gathered} \text{For any real number } a,\\ \sqrt[{\color{#FF0000}{n}}]{a^{\color{#FF0000}{n}}}= \begin{cases} \phantom{|}a\phantom{|} \text{ if } {\color{#FF0000}{n}} \text{ is odd}\\ |a| \text{ if } {\color{#FF0000}{n}} \text{ is even} \end{cases} \end{gathered}$ Since both radicals are real numbers and the roots are even, the expressions underneath the radicals are positive. Otherwise, the radicals would be imaginary. With this in mind, let's consider the possible values of the variables, $x$ and $y.$

• In the first radical, the index is even, the exponent of $x$ is odd, and the exponent of $y$ is even. Therefore, in order for this radical expression to result in a real number, $x$ must be positive.
• In the second radical, the index is even and the exponents of $x$ and $y$ are odd. Since $x$ must be positive, in order for this radical expression to result in a real number, $y$ also must be positive.
This means that both variables can only take positive values, so we do not need absolute value symbols.
$\sqrt[4]{81x^5y^4}\cdot \sqrt[4]{32x^3y}$
$\sqrt[4]{81x^5y^4\cdot32x^3y}$
$\sqrt[4]{81\cdot 32\cdot x^5\cdot x^3\cdot y^4 \cdot y}$
$\sqrt[4]{81\cdot 32\cdot x^8\cdot y^5}$
Next, let's simplify the radical expression by finding all of the perfect $4^\text{th}$ powers inside the radical.
$\sqrt[4]{81\cdot 32\cdot x^8\cdot y^5}$
Split into factors and write as powers
$\sqrt[4]{81\cdot 16\cdot 2\cdot x^{2\cdot 4}\cdot y^5}$
$\sqrt[4]{3^4\cdot 2^4\cdot 2 \cdot x^{2\cdot 4}\cdot y^5}$
$\sqrt[4]{3^4\cdot 2^4\cdot 2 \cdot \left(x^2\right)^4\cdot y^5}$
$\sqrt[4]{3^4\cdot 2^4\cdot 2 \cdot \left(x^2\right)^4\cdot y^{1+4}}$
$\sqrt[4]{3^4\cdot 2^4\cdot 2 \cdot \left(x^2\right)^4\cdot y\cdot y^4}$
$\sqrt[4]{3^4\cdot 2^4 \cdot \left(x^2\right)^4\cdot y^4\cdot2\cdot y}$
$\sqrt[4]{\left(6x^2y\right)^4\cdot 2\cdot y}$
$\sqrt[4]{\left(6x^2y\right)^4}\cdot \sqrt[4]{2\cdot y}$
$6x^2y\sqrt[4]{2\cdot y}$
$6x^2y\sqrt[4]{2y}$