To simplify the given expression [tex]\(\sqrt[3]{\frac{12 x^2}{16 y}}\)[/tex], we need to perform a few algebraic manipulations. Let's go through it step-by-step:
1. Simplify the coefficients:
The expression inside the cube root is:
[tex]\[
\frac{12 x^2}{16 y}
\][/tex]
We can simplify the fraction [tex]\(\frac{12}{16}\)[/tex]:
[tex]\[
\frac{12}{16} = \frac{3}{4}
\][/tex]
So the expression becomes:
[tex]\[
\frac{3 x^2}{4 y}
\][/tex]
2. Take the cube root:
Now, we need to take the cube root of the simplified fraction:
[tex]\[
\sqrt[3]{\frac{3 x^2}{4 y}}
\][/tex]
We can rewrite this as:
[tex]\[
\frac{\sqrt[3]{3 x^2}}{\sqrt[3]{4 y}}
\][/tex]
3. Simplify under the cube root:
We can separate the cube root for each component:
[tex]\[
\sqrt[3]{3 x^2} = 3^{1/3} x^{2/3}
\][/tex]
[tex]\[
\sqrt[3]{4 y} = 4^{1/3} y^{1/3}
\][/tex]
This transforms our expression into:
[tex]\[
\frac{3^{1/3} x^{2/3}}{4^{1/3} y^{1/3}}
\][/tex]
4. Match with given choices:
We need to see if this form matches any of the provided answers:
[tex]\[
\frac{2\left(\sqrt[3]{6 x^2 y^2}\right)}{y}
\][/tex]
[tex]\[
\frac{\sqrt[3]{12 x^2 y}}{2 y}
\][/tex]
[tex]\[
\frac{x(\sqrt[3]{3 y})}{2 y}
\][/tex]
[tex]\[
\frac{\sqrt[3]{6 x^2 y^2}}{2 y}
\][/tex]
Let's examine each choice:
1. [tex]\(\frac{2\left(\sqrt[3]{6 x^2 y^2}\right)}{y}\)[/tex]:
[tex]\[
= \frac{2 \cdot (6 x^2 y^2)^{1/3}}{y}
\][/tex]
This would not simplify correctly to our form.
2. [tex]\(\frac{\sqrt[3]{12 x^2 y}}{2 y}\)[/tex]
\\
This would be:
[tex]\[
= \frac{(12 x^2 y)^{1/3}}{2 y}
\][/tex]
This does not match our target form either.
3. [tex]\(\frac{x(\sqrt[3]{3 y})}{2 y}\)[/tex]:
[tex]\[
= \frac{x \cdot 3^{1/3} y^{1/3}}{2 y}
\][/tex]
Rearrange:
[tex]\[
= \frac{3^{1/3} x y^{1/3}}{2 y}
\][/tex]
This is close but still not matching.
4. [tex]\(\frac{\sqrt[3]{6 x^2 y^2}}{2 y}\)[/tex]:
[tex]\[
= \frac{(6 x^2 y^2)^{1/3}}{2 y}
\][/tex]
Simplify:
[tex]\[
= \frac{6^{1/3} x^{2/3} y^{2/3}}{2 y}
\][/tex]
Further simplify:
[tex]\[
= \frac{6^{1/3} x^{2/3} y^{2/3}}{2 y} = \frac{6^{1/3} x^{2/3} y^{2/3}}{2 y}
\][/tex]
Simplify again using the powers of [tex]\(y\)[/tex]:
[tex]\[
= \frac{6^{1/3} x^{2/3} y^{-1/3}}{2}
\][/tex]
So, we reach:
[tex]\(\=\sqrt[3]{6 x^{2}} y^{-2}} >2 \sqrt [ 2>} \)[/tex]
Hence the correct answer should be:
[tex]\(\frac{\sqrt[3]{6 x^2 y^2}}{2 y}\)[/tex]
