https://file.aiquickdraw.com/mathbot/file/1728692119564gcrb9427.jpg

The problem in the image asks to simplify the following expression:

\[
\frac{\left( a^{-7}b^{-1} \right)^{-1}}{(a^3b^3)^2}
\]

### Step-by-step solution:

1. **Simplify the numerator:**

   The expression in the numerator is \( \left( a^{-7}b^{-1} \right)^{-1} \). When an exponent is raised to a negative power, the base exponents change their signs. Hence:

   \[
   \left( a^{-7}b^{-1} \right)^{-1} = a^{7}b^{1} = a^7b
   \]

2. **Simplify the denominator:**

   The expression in the denominator is \( (a^3b^3)^2 \). When a power is raised to another power, multiply the exponents:

   \[
   (a^3b^3)^2 = a^{3 \times 2}b^{3 \times 2} = a^6b^6
   \]

3. **Combine the fraction:**

   Now we have the simplified expression:

   \[
   \frac{a^7b}{a^6b^6}
   \]

4. **Simplify the final expression:**

   Use the property of exponents to simplify:

   \[
   \frac{a^7}{a^6} = a^{7-6} = a^1 = a
   \]
   and
   \[
   \frac{b}{b^6} = b^{1-6} = b^{-5}
   \]

   Therefore, the final simplified result is:

   \[
   \boxed{ab^{-5}}
   \]

Let me know if you'd like more details or have any questions! Here are some related questions to consider:

1. How would the result change if the exponents in the numerator were positive?
2. What properties of exponents are used when simplifying fractions?
3. How does the negative exponent \( b^{-5} \) affect the final result?
4. Can the result be written differently using positive exponents?
5. What is the significance of raising powers to other powers in simplifying these expressions?

**Tip:** Always remember to apply exponent rules carefully when dealing with negative exponents or powers raised to other powers!

Sederhanakan bentuk pangkat (a^-7 * b^-1)^-1 / (a^3 * b^3)^2

Learn how to simplify the power expression (a^-7 * b^-1)^-1 / (a^3 * b^3)^2 by applying the power of a power rule and quotient of powers rule. Suitable for students in 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