The problem in the image asks you to simplify the expression $\left(-4a^{-4}b^5\right)^3$ and to write the answer using only positive exponents.

Let's solve this step by step:

Step 1: Apply the exponent to each term

The expression is $\left(-4a^{-4}b^5\right)^3$. We need to apply the exponent of 3 to each part inside the parentheses:

$(-4)^3 \cdot (a^{-4})^3 \cdot (b^5)^3$

Step 2: Simplify each part

• $(-4)^3 = -64$ (since $-4 \times -4 \times -4 = -64$)
• $(a^{-4})^3 = a^{-12}$ (use the power of a power rule: $(x^m)^n = x^{m \cdot n}$)
• $(b^5)^3 = b^{15}$ (again, apply the power of a power rule)

Now, we have:

$-64a^{-12}b^{15}$

Step 3: Convert negative exponents to positive

To write the answer using only positive exponents, we move $a^{-12}$ to the denominator:

$\frac{-64b^{15}}{a^{12}}$

Final Answer:

$\frac{-64b^{15}}{a^{12}}$

Would you like any more details or explanations on this?

Here are 5 related questions to explore:

1. What are the laws of exponents for multiplying terms with the same base?
2. How would the result change if one of the exponents were negative in the final answer?
3. How does the negative sign affect exponentiation in general?
4. Can you simplify $\left(3x^{-2}y^4\right)^2$ in a similar way?
5. What are the rules for converting negative exponents to positive in general?

Tip: Always remember that any term with a negative exponent can be rewritten as its reciprocal with a positive exponent.