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Rational Exponents and Radicals

Rational Exponents and Radicals 1.6 - Solution

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For any real number a,a, the radical expression ann\sqrt[n]{a^n} can be simplified as follows. ann={a if n is odda if n is even\begin{gathered} \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} Because the given root has an even index and the variables have even exponents, we will use the absolute value symbol to simplify the expression.
81g16h244\sqrt[4]{81g^{16}h^{24}}
34g16h244\sqrt[4]{3^4g^{16}h^{24}}
34g44h644\sqrt[4]{3^4g^{4\cdot 4}h^{6\cdot 4}}
34(g4)4(h6)44\sqrt[4]{3^4 \left(g^{4}\right)^4 \left(h^{6}\right)^4}
(3g4h6)44\sqrt[4]{\left(3g^{4}h^{6}\right)^4}
ann=a\sqrt[n]{a^n}=|a|
3g4h6\left|3g^{4}h^{6}\right|
Note that 33 is a positive number. Moreover, note that the expressions g4g^4 and h6h^6 have even exponents. Therefore, they are both positive. Since the product of three positive numbers is positive, the expression 3g4h63g^4h^6 is positive. 3g4h63g4h6\begin{gathered} \left|3g^{4}h^{6}\right| \quad \Leftrightarrow \quad 3g^4h^6 \end{gathered}