Questions: Simplify the expression using the properties of exponents. Expand any numerical portion of your answer and only include positive exponents. (-4x^3 / y^3)^3

Solution Steps

To simplify the given expression using the properties of exponents, we need to apply the power rule \((a^m)^n = a^{m \cdot n}\) to each component inside the parentheses. This involves raising both the numerical coefficient and each variable to the power of 3. Finally, ensure all exponents are positive.

We start with the expression

\[ \left(\frac{-4 x^{3}}{y^{3}}\right)^{3} \]

Using the power rule \((a^m)^n = a^{m \cdot n}\), we apply the exponent of 3 to both the coefficient and the variables.

The coefficient \(-4\) raised to the power of 3 is calculated as follows:

\[ (-4)^{3} = -64 \]

Next, we apply the exponent to the variables:

• For \(x^{3}\):

\[ (x^{3})^{3} = x^{3 \cdot 3} = x^{9} \]

• For \(y^{3}\):

\[ (y^{3})^{3} = y^{3 \cdot 3} = y^{9} \]

Combining all parts, we have:

\[ \frac{-64 x^{9}}{y^{9}} \]

Final Answer