Question

simplify.
$\left(-2x^{4}y^{-3}\
ight)^{3}$
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 can apply this to \((-2x^{4}y^{-3})^{3}\) as follows:
\((-2)^{3}(x^{4})^{3}(y^{-3})^{3}\)

Step2: Calculate each term

First, calculate \((-2)^{3}\): \((-2)^{3} = -8\)
Next, use the power of a power rule \((a^m)^n = a^{mn}\) for \((x^{4})^{3}\): \((x^{4})^{3}=x^{4\times3}=x^{12}\)
Then, use the power of a power rule for \((y^{-3})^{3}\): \((y^{-3})^{3}=y^{-3\times3}=y^{-9}\)
So now we have \(-8x^{12}y^{-9}\)

Step3: Convert negative exponents to positive

Recall that \(a^{-n}=\frac{1}{a^{n}}\), so \(y^{-9}=\frac{1}{y^{9}}\)
Substituting this back in, we get \(-8x^{12}\times\frac{1}{y^{9}}=\frac{-8x^{12}}{y^{9}}\)

Answer:

\(\frac{-8x^{12}}{y^{9}}\)