Certainly! Let's simplify the given expression step by step.
We start with the expression:
[tex]\[
\left(x^{-4} y^5\right)^{-3}
\][/tex]
### Step 1: Apply the Power of a Power Property
According to the power of a power property, [tex]\((a^m)^n = a^{m \cdot n}\)[/tex], we distribute the exponent [tex]\(-3\)[/tex] to both [tex]\(x^{-4}\)[/tex] and [tex]\(y^5\)[/tex]:
[tex]\[
\left(x^{-4}\right)^{-3} \cdot \left(y^5\right)^{-3}
\][/tex]
### Step 2: Multiply the Exponents
Now, we multiply the exponents:
For [tex]\(x^{-4}\)[/tex]:
[tex]\[
x^{-4 \cdot -3} = x^{12}
\][/tex]
For [tex]\(y^5\)[/tex]:
[tex]\[
y^{5 \cdot -3} = y^{-15}
\][/tex]
This gives us the expression:
[tex]\[
x^{12} \cdot y^{-15}
\][/tex]
### Step 3: Simplify to Eliminate Negative Exponents
An exponent rule tells us that [tex]\(a^{-m} = \frac{1}{a^m}\)[/tex]. Therefore, [tex]\(y^{-15}\)[/tex] can be written as [tex]\(\frac{1}{y^{15}}\)[/tex]:
[tex]\[
x^{12} \cdot \frac{1}{y^{15}} = \frac{x^{12}}{y^{15}}
\][/tex]
So, the simplified form of [tex]\(\left(x^{-4} y^5\right)^{-3}\)[/tex] with no negative exponents is:
[tex]\[
\boxed{\frac{x^{12}}{y^{15}}}
\][/tex]
