https://file.aiquickdraw.com/mathbot/file/1726791225407w65104vk.jpg

To simplify the given expression, let's break it down step-by-step. The expression is:

\[ y^7 \cdot (-2xy^2)^2 \cdot (-3x)^3 \]

### Step 1: Simplify \((-2xy^2)^2\)
To simplify \((-2xy^2)^2\), we raise each component to the power of 2:

\[
(-2)^2 = 4
\]
\[
(x)^2 = x^2
\]
\[
(y^2)^2 = y^4
\]

So, \((-2xy^2)^2 = 4x^2y^4\).

### Step 2: Simplify \((-3x)^3\)
To simplify \((-3x)^3\), we raise each component to the power of 3:

\[
(-3)^3 = -27
\]
\[
(x)^3 = x^3
\]

So, \((-3x)^3 = -27x^3\).

### Step 3: Combine All Terms
Now, multiply all the simplified components together:

\[
y^7 \cdot 4x^2y^4 \cdot (-27x^3)
\]

Combine the coefficients:

\[
4 \cdot (-27) = -108
\]

Combine the powers of \( x \) and \( y \):

\[
x^2 \cdot x^3 = x^{2+3} = x^5
\]
\[
y^7 \cdot y^4 = y^{7+4} = y^{11}
\]

### Final Simplified Expression
\[
-108x^5y^{11}
\]

This is the simplified form of the given expression.

Do you have any questions or want more details? Here are five related questions:

1. How do you multiply terms with the same base?
2. What is the general rule for raising a product to a power?
3. How do you handle negative signs in exponents?
4. Can you simplify \( (x^2 y^3)^4 \) and explain the process?
5. What is the significance of the coefficients in the final result?

**Tip:** Always simplify each component separately before multiplying them together for a more straightforward approach.

This guide walks through simplifying the expression y^7 * (-2xy^2)^2 * (-3x)^3 by applying laws of exponents and algebraic simplifications. Suitable for high school 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