Use the rules of exponents to simplify the following expression. Assume that all variables are nonzero.

Question

left parenthesis 4 x Superscript negative 2 Baseline y Superscript 5 Baseline right parenthesis Superscript negative 3

Answer 1

To simplify the expression \( \left( 4 x^{-2} y^{5} \right)^{-3} \), we can use the rules of exponents.

1. Apply the power of a product rule: \( (ab)^n = a^n b^n \)

   Therefore, we can distribute the exponent \(-3\) to each factor: \[ (4)^{-3} (x^{-2})^{-3} (y^{5})^{-3} \]

2. Simplify each term:

   • For \( (4)^{-3} \): \[ (4)^{-3} = \frac{1}{4^3} = \frac{1}{64} \]
   • For \( (x^{-2})^{-3} \): \[ (x^{-2})^{-3} = x^{-2 \cdot -3} = x^{6} \]
   • For \( (y^{5})^{-3} \): \[ (y^{5})^{-3} = y^{5 \cdot -3} = y^{-15} \]

3. Combine the results: Putting it all together, we have: \[ \left( 4 x^{-2} y^{5} \right)^{-3} = \frac{1}{64} x^{6} y^{-15} \]

4. Express the negative exponent in the denominator: Rewrite \( y^{-15} \) as \( \frac{1}{y^{15}} \): \[ = \frac{x^{6}}{64 y^{15}} \]

Thus, the simplified expression is: \[ \frac{x^{6}}{64 y^{15}} \]