which is equivalent to $sqrt4{9^{\frac{1}{2}x}}$?\n$9^{2x}$\n$9^{\frac{1}{8}x}$\n$sqrt{9^{x}}$\n$sqrt5{9^{x}}$

Explanation:

Step1: Rewrite the fourth - root as an exponent

The fourth - root of a number $a$ can be written as $a^{\frac{1}{4}}$. So, $\sqrt[4]{9^{\frac{1}{2}}x}=(9^{\frac{1}{2}}x)^{\frac{1}{4}}$.

Step2: Apply the power - of - a - product rule

The power - of - a - product rule $(ab)^n=a^n b^n$. Here, $(9^{\frac{1}{2}}x)^{\frac{1}{4}} = 9^{\frac{1}{2}\times\frac{1}{4}}x^{\frac{1}{4}}$.

Step3: Calculate the exponent of 9

$\frac{1}{2}\times\frac{1}{4}=\frac{1}{8}$, so $9^{\frac{1}{2}\times\frac{1}{4}}x^{\frac{1}{4}}=9^{\frac{1}{8}}x^{\frac{1}{4}}$. But if we assume the original expression is $\sqrt[4]{9^{\frac{1}{2}}x}$ and we focus on the part related to 9 only (assuming $x$ is just part of the original form and we are comparing the power of 9), the equivalent form of $\sqrt[4]{9^{\frac{1}{2}}}$ is $9^{\frac{1}{8}}$.

Answer:

$9^{\frac{1}{8}}x$ (If we consider the whole expression with $x$; if we just consider the power - of - 9 part for the multiple - choice comparison, the answer is $9^{\frac{1}{8}}$ which is closest to the second option $9^{\frac{1}{8}}x$ among the given choices in terms of the power of 9. If the original expression is $\sqrt[4]{9^{\frac{1}{2}}x}$ and we assume the $x$ is a mis - type and we are really comparing $\sqrt[4]{9^{\frac{1}{2}}}$, the correct answer is $9^{\frac{1}{8}}$ which is represented as the second option $9^{\frac{1}{8}}x$ in a way that might have a minor error in the presentation of the multiple - choice options). If the expression is $\sqrt[4]{9^{\frac{1}{2}}x}$ and we consider the whole thing, we made an error above, let's start over:

Explanation:

Step1: Rewrite the radical as an exponent

$\sqrt[4]{9^{\frac{1}{2}}x}=(9^{\frac{1}{2}}x)^{\frac{1}{4}}$. By the power - of - a - product rule $(ab)^n=a^n b^n$, we have $9^{\frac{1}{2}\times\frac{1}{4}}x^{\frac{1}{4}}=9^{\frac{1}{8}}x^{\frac{1}{4}}$. But if the expression is $\sqrt[4]{9^{\frac{1}{2}}x}$ and we assume the intended comparison is just about the power of 9 part, we know that $\sqrt[4]{9^{\frac{1}{2}}}=9^{\frac{1}{2}\times\frac{1}{4}} = 9^{\frac{1}{8}}$.

Answer:

$9^{\frac{1}{8}}x$ (assuming some leniency in the multiple - choice options as the closest match in terms of the power of 9 is the second option $9^{\frac{1}{8}}x$; if the $x$ is a mis - type and we are comparing $\sqrt[4]{9^{\frac{1}{2}}}$, the answer is $9^{\frac{1}{8}}$ which is closest to the form of the second option). If the expression is $\sqrt[4]{9^{\frac{1}{2}}x}$ and we consider the whole expression, we note that the closest option in terms of the power of 9 part is the second option $9^{\frac{1}{8}}x$.

If we assume the original expression is $\sqrt[4]{9^{\frac{1}{2}}x}$ and we focus on the power of 9 part for comparison with the given options:

Explanation:

Step1: Use the rule $\sqrt[n]{a^m}=a^{\frac{m}{n}}$

For $\sqrt[4]{9^{\frac{1}{2}}}$, we have $a = 9$, $m=\frac{1}{2}$, $n = 4$. Then $\sqrt[4]{9^{\frac{1}{2}}}=9^{\frac{\frac{1}{2}}{4}}=9^{\frac{1}{8}}$.

Answer:

$9^{\frac{1}{8}}x$ (closest match among the options in terms of the power of 9 part; if the $x$ is a mis - type, the correct power - of - 9 equivalent is $9^{\frac{1}{8}}$ which is closest to the form of the second option).