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Which expression is equivalent to [tex]\left(32^2 x^4 y^{12}\right)^{\frac{1}{5}}[/tex]?

A. [tex]4 y^2 \sqrt[5]{x^2 y^4}[/tex]

B. [tex]4 y^2 \sqrt[5]{x^4 y^2}[/tex]

C. [tex]2 y^2 \sqrt[5]{x^2 y^4}[/tex]

D. [tex]2 y^2 \sqrt[5]{x^4 y^2}[/tex]


Sagot :

To find the equivalent expression to [tex]\(\left(32^2 x^4 y^{12}\right)^{\frac{1}{5}}\)[/tex], let's simplify the given expression step by step.

1. Given Expression:
[tex]\[ \left(32^2 x^4 y^{12}\right)^{\frac{1}{5}} \][/tex]

2. Simplify the Inside:
- Calculate [tex]\(32^2\)[/tex]:
[tex]\[ 32^2 = 1024 \][/tex]
- The expression becomes:
[tex]\[ \left(1024 x^4 y^{12}\right)^{\frac{1}{5}} \][/tex]

3. Apply the Exponent [tex]\(\frac{1}{5}\)[/tex] to Each Term:
- Rewrite the expression:
[tex]\[ \left(1024\right)^{\frac{1}{5}} \left(x^4\right)^{\frac{1}{5}} \left(y^{12}\right)^{\frac{1}{5}} \][/tex]

4. Simplify Each Part:
- For [tex]\(1024^{\frac{1}{5}}\)[/tex]:
[tex]\[ 1024^{\frac{1}{5}} = \left(2^{10}\right)^{\frac{1}{5}} = 2^{2} = 4 \][/tex]
- For [tex]\(\left(x^4\right)^{\frac{1}{5}}\)[/tex]:
[tex]\[ \left(x^4\right)^{\frac{1}{5}} = x^{\frac{4}{5}} \][/tex]
- For [tex]\(\left(y^{12}\right)^{\frac{1}{5}}\)[/tex]:
[tex]\[ \left(y^{12}\right)^{\frac{1}{5}} = y^{\frac{12}{5}} \][/tex]

5. Combine the Simplified Parts:
- The expression now looks like:
[tex]\[ 4 \cdot x^{\frac{4}{5}} \cdot y^{\frac{12}{5}} \][/tex]

6. Rewrite the Expression in a More Common Form:
- Notice that [tex]\( \frac{12}{5} = 2 + \frac{2}{5} \)[/tex]:
[tex]\[ y^{\frac{12}{5}} = y^2 \cdot y^{\frac{2}{5}} \][/tex]

- The expression becomes:
[tex]\[ 4 \cdot x^{\frac{4}{5}} \cdot y^2 \cdot y^{\frac{2}{5}} \Rightarrow 4 y^2 \cdot \left(x^{\frac{4}{5}} y^{\frac{2}{5}}\right) \][/tex]

- Recognize a common term:
[tex]\[ x^{\frac{4}{5}} y^{\frac{2}{5}} = \left(x^4 y^2\right)^{\frac{1}{5}} \][/tex]

Finally, the expression simplifies to:
[tex]\[ 4 y^2 \sqrt[5]{x^4 y^2} \][/tex]

So, the correct answer is:
[tex]\[ 4 y^2 \sqrt[5]{x^4 y^2} \][/tex]

Thus, the correct option is:
[tex]\[ \boxed{4 y^2 \sqrt[5]{x^4 y^2}} \][/tex]