To solve the problem, we need to simplify the expression:
\[
\left( \frac{-54x^9}{y^4} \right)^{\frac{2}{3}}
\]
Step 1: Apply the exponent to both the numerator and the denominator
The expression inside the parentheses is a fraction, so we can apply the exponent $\frac{2}{3}$ to both the numerator and the denominator separately:
\[
\left( \frac{-54x^9}{y^4} \right)^{\frac{2}{3}} = \frac{\left( -54x^9 \right)^{\frac{2}{3}}}{\left( y^4 \right)^{\frac{2}{3}}}
\]
Step 2: Simplify the numerator $\left( -54x^9 \right)^{\frac{2}{3}}$
We can break this down into two parts: the coefficient $-54$ and the variable part $x^9$.
#### Simplify the coefficient $(-54)^{\frac{2}{3}}$:
First, express $-54$ as a product of its prime factors:
\[
-54 = -1 \cdot 2 \cdot 3^3
\]
Now, apply the exponent $\frac{2}{3}$:
\[
(-54)^{\frac{2}{3}} = \left( -1 \cdot 2 \cdot 3^3 \right)^{\frac{2}{3}} = (-1)^{\frac{2}{3}} \cdot 2^{\frac{2}{3}} \cdot (3^3)^{\frac{2}{3}}
\]
- $(-1)^{\frac{2}{3}} = 1$ because raising $-1$ to an even power results in $1$.
- $(3^3)^{\frac{2}{3}} = 3^{3 \cdot \frac{2}{3}} = 3^2 = 9$.
- $2^{\frac{2}{3}}$ remains as it is.
So:
\[
(-54)^{\frac{2}{3}} = 1 \cdot 2^{\frac{2}{3}} \cdot 9 = 9 \cdot 2^{\frac{2}{3}}
\]
#### Simplify the variable part $(x^9)^{\frac{2}{3}}$:
\[
(x^9)^{\frac{2}{3}} = x^{9 \cdot \frac{2}{3}} = x^6
\]
Combining these, the numerator becomes:
\[
\left( -54x^9 \right)^{\frac{2}{3}} = 9 \cdot 2^{\frac{2}{3}} \cdot x^6
\]
Step 3: Simplify the denominator $\left( y^4 \right)^{\frac{2}{3}}$
\[
\left( y^4 \right)^{\frac{2}{3}} = y^{4 \cdot \frac{2}{3}} = y^{\frac{8}{3}}
\]
Step 4: Combine the simplified numerator and denominator
\[
\left( \frac{-54x^9}{y^4} \right)^{\frac{2}{3}} = \frac{9 \cdot 2^{\frac{2}{3}} \cdot x^6}{y^{\frac{8}{3}}}
\]
Step 5: Express $y^{\frac{8}{3}}$ in a more familiar form
Recall that $y^{\frac{8}{3}} = y^{2 + \frac{2}{3}} = y^2 \cdot y^{\frac{2}{3}}$. Thus:
\[
\frac{9 \cdot 2^{\frac{2}{3}} \cdot x^6}{y^{\frac{8}{3}}} = \frac{9 \cdot 2^{\frac{2}{3}} \cdot x^6}{y^2 \cdot y^{\frac{2}{3}}}
\]
Step 6: Write $2^{\frac{2}{3}}$ and $y^{\frac{2}{3}}$ using cube roots
\[
2^{\frac{2}{3}} = \sqrt[3]{2^2} = \sqrt[3]{4}
\]
\[
y^{\frac{2}{3}} = \sqrt[3]{y^2}
\]
Substitute these back into the expression:
\[
\frac{9 \cdot 2^{\frac{2}{3}} \cdot x^6}{y^2 \cdot y^{\frac{2}{3}}} = \frac{9 \cdot \sqrt[3]{4} \cdot x^6}{y^2 \cdot \sqrt[3]{y^2}}
\]
Final Answer:
\[
\boxed{\frac{9x^6 \sqrt[3]{4}}{y^2 \sqrt[3]{y^2}}}
\]
This matches option (2). Therefore, the correct answer is:
\[
\boxed{2}
\]
Parent Tip: Review the logic above to help your child master the concept of math problem algebra 2 equations.
