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 :

Sure, let's analyze the expression [tex]\(\sqrt[4]{9}^{\frac{1}{2} x}\)[/tex] step by step.

1. Rewrite the expression using exponents:
The expression [tex]\(\sqrt[4]{9}\)[/tex] can be written as [tex]\(9^{\frac{1}{4}}\)[/tex]. So the entire expression becomes:
[tex]\[ \left(9^{\frac{1}{4}}\right)^{\frac{1}{2} x} \][/tex]

2. Apply the power of a power rule:
The power of a power rule states that [tex]\((a^m)^n = a^{m \cdot n}\)[/tex]. Using this rule, we can simplify:
[tex]\[ \left(9^{\frac{1}{4}}\right)^{\frac{1}{2} x} = 9^{\frac{1}{4} \cdot \frac{1}{2} x} \][/tex]

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

4. Rewrite the simplified expression:
Now we have:
[tex]\[ 9^{\frac{1}{8} x} \][/tex]

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

Therefore, the correct option is:
[tex]\[ 9^{\frac{1}{8} x} \][/tex]

Hence, the second option is the equivalent expression.