Question

simplify.
$\left(-3x^{-6}y^{5}\
ight)^{4}$
write your answer using only positive exponents.

Explanation:

Step1: Apply the power of a product rule

The power of a product rule states that \((ab)^n = a^n b^n\). So, we apply this to \((-3x^{-6}y^{5})^{4}\):
\[
(-3)^{4}(x^{-6})^{4}(y^{5})^{4}
\]

Step2: Calculate each term

• Calculate \((-3)^{4}\): \((-3)^{4}=(-3)\times(-3)\times(-3)\times(-3) = 81\)
• For \((x^{-6})^{4}\), use the power of a power rule \((a^m)^n=a^{mn}\). So, \((x^{-6})^{4}=x^{-6\times4}=x^{-24}\)
• For \((y^{5})^{4}\), use the power of a power rule: \((y^{5})^{4}=y^{5\times4}=y^{20}\)

So now we have \(81x^{-24}y^{20}\)

Step3: Convert negative exponents to positive

Recall that \(a^{-n}=\frac{1}{a^{n}}\), so \(x^{-24}=\frac{1}{x^{24}}\). But we want to write with only positive exponents, so we can rewrite \(x^{-24}\) in the denominator (or use the rule \(a^{-n}=\frac{1}{a^{n}}\) to make the exponent positive). So, \(81x^{-24}y^{20}=\frac{81y^{20}}{x^{24}}\)

Answer:

\(\frac{81y^{20}}{x^{24}}\)