https://file.aiquickdraw.com/mathbot/file/1726332140609gj6hbcs1.jpg

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.

Simplificar \((-4a^{-4}b^5)^3\) escribiendo la respuesta utilizando sólo exponentes positivos.

Learn how to simplify the expression \((-4a^{-4}b^5)^3\) by applying exponent rules, including the power of a power rule and converting negative exponents to positive. Detailed steps and explanation provided. Suitable for Grades 9-10.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation