Answer :

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]