Question

use the following image to answer the question.
question 46
write the following expression using rational exponents.
\sqrt6{(3a)^7}
select one
\bigcirc 3a^{\frac{6}{7}}
\bigcirc 3a^{\frac{7}{6}}
\bigcirc (3a)^{\frac{7}{6}}
\bigcirc (3a)^{\frac{6}{7}}

Explanation:

Step1: Recall the radical to exponent rule

The formula for converting a radical to a rational exponent is $\sqrt[n]{x^m} = x^{\frac{m}{n}}$. Here, the radical is $\sqrt[6]{(3a)^7}$, so $n = 6$ and $m = 7$, and $x=(3a)$.

Step2: Apply the rule

Substitute into the formula: $\sqrt[6]{(3a)^7}=(3a)^{\frac{7}{6}}$. Wait, no, wait. Wait, the general rule is $\sqrt[n]{x}=x^{\frac{1}{n}}$, and $\sqrt[n]{x^m}=x^{\frac{m}{n}}$. So for $\sqrt[6]{(3a)^7}$, it's $(3a)^{\frac{7}{6}}$? Wait, but let's check the options. Wait, the options have $(3a)^{\frac{7}{6}}$? Wait, the third option? Wait no, the options are: first: $3a^{\frac{6}{7}}$, second: $3a^{\frac{7}{6}}$, third: $(3a)^{\frac{7}{6}}$, fourth: $(3a)^{\frac{6}{7}}$. Wait, wait, the original expression is $\sqrt[6]{(3a)^7}$. So using the rule $\sqrt[n]{x^m}=x^{\frac{m}{n}}$, so here $x=(3a)$, $m = 7$, $n = 6$. So it's $(3a)^{\frac{7}{6}}$, which is the third option? Wait, the third option is $(3a)^{\frac{7}{6}}$? Wait, the user's image: the options are:

1. $3a^{\frac{6}{7}}$

1. $3a^{\frac{7}{6}}$

1. $(3a)^{\frac{7}{6}}$

1. $(3a)^{\frac{6}{7}}$

Wait, so the correct one is the third option: $(3a)^{\frac{7}{6}}$. Wait, let's recheck. The radical is sixth root of (3a) to the seventh power. So $\sqrt[6]{(3a)^7}=(3a)^{\frac{7}{6}}$, which is option C (the third option, labeled as $(3a)^{\frac{7}{6}}$). Wait, but in the image, the third option is selected? Wait no, the blue dot is on the fourth option? Wait, no, maybe I misread. Wait, the fourth option is $(3a)^{\frac{6}{7}}$, third is $(3a)^{\frac{7}{6}}$. Wait, no, let's do the rule again. The formula is $\sqrt[n]{x^m}=x^{m/n}$. So for $\sqrt[6]{(3a)^7}$, it's $(3a)^{7/6}$. So that's the third option. Wait, maybe the user made a typo, but according to the rule, the correct expression is $(3a)^{\frac{7}{6}}$, which is the third option (the one with $(3a)^{\frac{7}{6}}$).

Answer:

C. $(3a)^{\frac{7}{6}}$ (assuming the third option is labeled as such, with the expression $(3a)^{\frac{7}{6}}$)