https://file.aiquickdraw.com/mathbot/file/1725515378985bt7adzia.jpg

Let's solve the given expression step by step.

The given expression is:

\[
\frac{\left( 2x^{-3} y^4 \right)^3 \left( \frac{x^2 y^{-3}}{4x y^5} \right)^{-2}}{16x^{-6} y^{12}}
\]

### Step 1: Simplify \(\left( 2x^{-3} y^4 \right)^3\)
Apply the power rule \((a^m b^n)^p = a^{mp} b^{np}\):

\[
\left( 2x^{-3} y^4 \right)^3 = 2^3 x^{-9} y^{12} = 8x^{-9} y^{12}
\]

### Step 2: Simplify \(\left( \frac{x^2 y^{-3}}{4x y^5} \right)^{-2}\)
First, simplify the fraction inside the parentheses:

\[
\frac{x^2 y^{-3}}{4x y^5} = \frac{x^2}{4x} \cdot \frac{y^{-3}}{y^5} = \frac{x^1}{4} \cdot y^{-8} = \frac{x y^{-8}}{4}
\]

Now, apply the exponent of \(-2\) to this fraction:

\[
\left( \frac{x y^{-8}}{4} \right)^{-2} = \frac{4^2}{(x y^{-8})^2} = \frac{16}{x^2 y^{-16}} = \frac{16 y^{16}}{x^2}
\]

### Step 3: Multiply the results from Step 1 and Step 2
Multiply the two simplified terms:

\[
8x^{-9} y^{12} \cdot \frac{16 y^{16}}{x^2} = \frac{8 \cdot 16 \cdot y^{12} \cdot y^{16}}{x^{-9} \cdot x^2} = \frac{128 y^{28}}{x^{-7}}
\]

Since \(x^{-9} \cdot x^2 = x^{-7}\), and simplifying \(x^{-7}\) moves it to the numerator as \(x^7\):

\[
128 x^7 y^{28}
\]

### Step 4: Divide by the denominator \(16x^{-6} y^{12}\)
Now divide the result from Step 3 by \(16x^{-6} y^{12}\):

\[
\frac{128 x^7 y^{28}}{16 x^{-6} y^{12}} = \frac{128}{16} \cdot \frac{x^7}{x^{-6}} \cdot \frac{y^{28}}{y^{12}} = 8 \cdot x^{7 - (-6)} \cdot y^{28 - 12}
\]

This simplifies to:

\[
8 x^{13} y^{16}
\]

### Final Answer:
\[
8 x^{13} y^{16}
\]

Let me know if you would like more details or have any other questions!

Here are five follow-up questions related to this topic:
1. How do you simplify expressions involving negative exponents?
2. Can you apply the power of a quotient rule to another example?
3. How does multiplying terms with exponents work?
4. What happens if a base is raised to a negative power and divided?
5. How does exponentiation affect both the numerator and denominator in a fraction?

**Tip**: Always simplify expressions inside parentheses before applying any exponent rules!

Simplify the expression: ((2x^-3 * y^4)^3 * (x^2 * y^-3)/(4x * y^5))^-2 / (16x^-6 * y^12)

Learn how to simplify complex algebraic expressions involving negative exponents and fractions. This problem uses exponentiation rules and the power of a quotient rule. Suitable for students in Grades 9-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation