https://r2.erweima.ai/mathbot/image/1725786718526fl3y8bvn.jpg

The expression in the image is:

\[
\left( \frac{3^{-1} \cdot a^{-4} \cdot b^5}{2a^2 \cdot b^{-2}} \right)^2 \cdot \left( \frac{-b}{3a^4} \right)^2
\]

Let's simplify this step by step.

### Step 1: Simplify the first fraction inside the first parentheses
We start by simplifying the terms in the first fraction:

\[
\frac{3^{-1} \cdot a^{-4} \cdot b^5}{2a^2 \cdot b^{-2}}
\]

Using the property of exponents, we can combine like terms:

- For the base \( a \): \( a^{-4} \div a^2 = a^{-6} \)
- For the base \( b \): \( b^5 \div b^{-2} = b^7 \)
- For the constant \( 3^{-1} \): \( 3^{-1} = \frac{1}{3} \)

Thus, the fraction simplifies to:

\[
\frac{\frac{1}{3} \cdot a^{-6} \cdot b^7}{2} = \frac{a^{-6} \cdot b^7}{6}
\]

### Step 2: Square the simplified fraction
Now, square the simplified fraction:

\[
\left( \frac{a^{-6} \cdot b^7}{6} \right)^2 = \frac{a^{-12} \cdot b^{14}}{36}
\]

### Step 3: Simplify the second fraction inside the second parentheses
Now, simplify the second fraction:

\[
\left( \frac{-b}{3a^4} \right)^2
\]

When squaring the fraction, we get:

\[
\left( \frac{-b}{3a^4} \right)^2 = \frac{b^2}{9a^8}
\]

### Step 4: Multiply the two squared fractions
Now, multiply the results from Step 2 and Step 3:

\[
\frac{a^{-12} \cdot b^{14}}{36} \cdot \frac{b^2}{9a^8} = \frac{a^{-12} \cdot b^{14} \cdot b^2}{36 \cdot 9 \cdot a^8} = \frac{a^{-12} \cdot b^{16}}{324 \cdot a^8}
\]

### Step 5: Simplify the final expression
Simplify the powers of \( a \):

\[
\frac{a^{-12} \cdot b^{16}}{324 \cdot a^8} = \frac{b^{16}}{324 \cdot a^{20}}
\]

### Final Answer:

\[
\boxed{\frac{b^{16}}{324a^{20}}}
\]

Let me know if you need further details or explanations!

### Related Questions:
1. How do you simplify fractional exponents?
2. What are the rules for multiplying exponents with the same base?
3. How do negative exponents affect terms in fractions?
4. Can you explain why squaring a negative term results in a positive term?
5. What other algebraic simplifications can help in fraction problems?

### Tip:
Always group terms with the same base when simplifying complex expressions to avoid mistakes.

Simplify the expression: ((3^-1 * a^-4 * b^5) / (2a^2 * b^-2))^2 * ((-b) / (3a^4))^2

This guide simplifies the algebraic expression ((3^-1 * a^-4 * b^5) / (2a^2 * b^-2))^2 * ((-b) / (3a^4))^2 step-by-step, involving exponent rules, fraction simplification, and multiplication of powers. Ideal for students in Grades 10-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation