Which is equivalent to [tex]\sqrt[4]{9}^{\frac{1}{2} x}[/tex]?

A. [tex]9^{2 x}[/tex]

B. [tex]9^{\frac{1}{8} x}[/tex]

C. [tex]\sqrt{9}^x[/tex]

D. [tex]\sqrt[5]{9}^x[/tex]



Answer :

To find the expression equivalent to [tex]\(\sqrt[4]{9}^{\frac{1}{2} x}\)[/tex], we will simplify it step by step.

1. Begin with the given expression:
[tex]\[ \sqrt[4]{9}^{\frac{1}{2} x} \][/tex]

2. Rewrite [tex]\(\sqrt[4]{9}\)[/tex] as an exponent. The fourth root of 9 can be written as:
[tex]\[ 9^{\frac{1}{4}} \][/tex]

So the original expression becomes:
[tex]\[ \left(9^{\frac{1}{4}}\right)^{\frac{1}{2} x} \][/tex]

3. Use the rule of exponents [tex]\((a^m)^n = a^{m \cdot n}\)[/tex] to combine the exponents:
[tex]\[ 9^{\left(\frac{1}{4} \cdot \frac{1}{2} x\right)} \][/tex]

4. Multiply the exponents:
[tex]\[ \frac{1}{4} \cdot \frac{1}{2} x = \frac{1}{8} x \][/tex]

5. This simplifies the expression to:
[tex]\[ 9^{\frac{1}{8} x} \][/tex]

Therefore, the expression equivalent to [tex]\(\sqrt[4]{9}^{\frac{1}{2} x}\)[/tex] is [tex]\(9^{\frac{1}{8} x}\)[/tex].

The correct choice is:
[tex]\[ 9^{\frac{1}{8} x} \][/tex]