when $9^{\frac{2}{3}}$ is written in simplest radical form, which value remains under the radical?\n3\n6\n9\n…
when $9^{\frac{2}{3}}$ is written in simplest radical form, which value remains under the radical?\n3\n6\n9\n27
Answer
Explanation:
Step1: Rewrite the base as a power
We know that $9 = 3^2$. So, $9^{\frac{2}{3}}=(3^2)^{\frac{2}{3}}$.
Step2: Apply power - of - a - power rule
According to the rule $(a^m)^n=a^{mn}$, then $(3^2)^{\frac{2}{3}} = 3^{2\times\frac{2}{3}}=3^{\frac{4}{3}}$.
Step3: Rewrite as a radical
The fractional - exponent rule $a^{\frac{m}{n}}=\sqrt[n]{a^m}$ gives $3^{\frac{4}{3}}=\sqrt[3]{3^4}$.
Step4: Simplify the radical
$3^4 = 81$, and $\sqrt[3]{3^4}=\sqrt[3]{81}=\sqrt[3]{27\times3}=3\sqrt[3]{3}$.
Answer:
3