To simplify the expression
[tex]\[
\left(-4 a^5 b^{-3}\right)^{-3}
\][/tex]
follow these steps:
1. Rewrite the Expression Inside the Parentheses:
Start by understanding the base expression
[tex]\[
-4 a^5 b^{-3}
\][/tex].
2. Apply the Power to Each Factor:
When raising a product to a power, raise each factor to that power. Thus,
[tex]\[
\left(-4 a^5 b^{-3}\right)^{-3} = (-4)^{-3} \cdot (a^5)^{-3} \cdot (b^{-3})^{-3}.
\][/tex]
3. Simplify Each Factor Individually:
- For the constant [tex]\(-4\)[/tex]:
[tex]\[
(-4)^{-3} = \frac{1}{(-4)^3} = \frac{1}{-64}.
\][/tex]
- For the [tex]\(a\)[/tex]-term:
[tex]\[
(a^5)^{-3} = a^{5 \cdot (-3)} = a^{-15}.
\][/tex]
- For the [tex]\(b\)[/tex]-term:
[tex]\[
(b^{-3})^{-3} = b^{(-3) \cdot (-3)} = b^9.
\][/tex]
4. Combine the Simplified Factors:
Now, multiply the simplified factors together:
[tex]\[
\frac{1}{-64} \cdot a^{-15} \cdot b^9.
\][/tex]
5. Rewrite the Expression in a Standard Form:
Finally, rewrite the expression so that negative exponents are moved to the denominator:
[tex]\[
\frac{b^9}{64 a^{15}}.
\][/tex]
Therefore, the simplified form of the expression
[tex]\[
\left(-4 a^5 b^{-3}\right)^{-3}
\][/tex]
is
[tex]\[
\frac{b^9}{64 a^{15}}.
\][/tex]
