Question

simplify. express your answer using exponents.
$(g^{-2}h^{5})^{3}$

Explanation:

Step1: Apply the power of a product rule

The power of a product rule states that \((ab)^n = a^n b^n\). So, for \((g^{-2}h^{5})^{3}\), we can apply this rule to get \( (g^{-2})^{3}(h^{5})^{3} \).

Step2: Apply the power of a power rule

The power of a power rule states that \((a^m)^n = a^{mn}\). For \( (g^{-2})^{3} \), we multiply the exponents: \(-2\times3 = -6\), so this becomes \( g^{-6} \). For \( (h^{5})^{3} \), we multiply the exponents: \(5\times3 = 15\), so this becomes \( h^{15} \).

Step3: Combine the results

Putting it together, we have \( g^{-6}h^{15} \). If we want to express the negative exponent as a positive exponent in the denominator, it can also be written as \( \frac{h^{15}}{g^{6}} \), but since the problem just says to express using exponents (and negative exponents are still exponents), \( g^{-6}h^{15} \) is also acceptable. However, usually, we can leave it with the negative exponent or convert it. But following the rules of exponents, the simplified form is \( g^{-6}h^{15} \) or \( \frac{h^{15}}{g^{6}} \).

Answer:

\( g^{-6}h^{15} \) (or \( \frac{h^{15}}{g^{6}} \))