https://file.aiquickdraw.com/mathbot/file/1730989863003gjryfafp.jpg

To solve the given problem, we need to simplify the expression:

\[
\frac{(8x^3y^2)^2}{(2x)^5y}
\]

Let's go through it step-by-step.

### Step 1: Simplify the Numerator
The numerator is \((8x^3y^2)^2\). Apply the power of a product rule \((ab)^n = a^n b^n\), which gives:

\[
(8x^3y^2)^2 = 8^2 \cdot (x^3)^2 \cdot (y^2)^2
\]

Calculate each term:
- \(8^2 = 64\)
- \((x^3)^2 = x^{3 \times 2} = x^6\)
- \((y^2)^2 = y^{2 \times 2} = y^4\)

So, the numerator becomes:

\[
64x^6y^4
\]

### Step 2: Simplify the Denominator
The denominator is \((2x)^5 y\). Again, apply the power of a product rule:

\[
(2x)^5 = 2^5 \cdot x^5
\]

Calculate each term:
- \(2^5 = 32\)
- \(x^5\) remains as it is.

So, the denominator becomes:

\[
32x^5y
\]

### Step 3: Divide the Numerator by the Denominator
Now we have:

\[
\frac{64x^6y^4}{32x^5y}
\]

Divide each term separately:
- **Constants**: \(\frac{64}{32} = 2\)
- **\(x\)-terms**: \(\frac{x^6}{x^5} = x^{6-5} = x\)
- **\(y\)-terms**: \(\frac{y^4}{y} = y^{4-1} = y^3\)

Putting it all together, the expression simplifies to:

\[
2xy^3
\]

### Final Answer
\[
\boxed{2xy^3}
\]

Would you like further details or have any questions?

Here are some related questions for further practice:
1. How would you simplify \(\frac{(3x^4y)^3}{(x^2y^2)^2}\)?
2. What is the result of \((4x^2y^3)^3 / (2x^5y)\)?
3. How do you simplify \(\frac{(10a^2b)^3}{(5a)^4b^2}\)?
4. Simplify \(\frac{(7x^2y)^2}{(x^3y)^3}\).
5. What steps would you take to simplify \(\frac{(5x^4y^3)^2}{(x^6y)}\)?

**Tip:** When simplifying expressions with exponents, remember to carefully apply the power of a product and the division of powers rules to avoid common mistakes.

Simplify the expression \(\frac{(8x^3y^2)^2}{(2x)^5y}\)

Learn how to simplify the algebraic expression \(\frac{(8x^3y^2)^2}{(2x)^5y}\) using exponent rules. This step-by-step solution covers exponentiation, the power of a product, and division of exponents, ideal for students in Grades 9-11.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation