Let's simplify the expression \( 2\sqrt{32x^2} + 8\sqrt{512x^2} \).

### Step 1: Simplify the square roots
First, let's simplify each square root individually.

#### Simplifying \( \sqrt{32x^2} \):
\[
\sqrt{32x^2} = \sqrt{32} \cdot \sqrt{x^2}
\]
Since \( \sqrt{x^2} = |x| \), we have:
\[
\sqrt{32} = \sqrt{16 \cdot 2} = 4\sqrt{2}
\]
Thus:
\[
\sqrt{32x^2} = 4\sqrt{2} \cdot |x| = 4|x|\sqrt{2}
\]
So:
\[
2\sqrt{32x^2} = 2 \cdot 4|x|\sqrt{2} = 8|x|\sqrt{2}
\]

#### Simplifying \( \sqrt{512x^2} \):
\[
\sqrt{512x^2} = \sqrt{512} \cdot \sqrt{x^2}
\]
Again, \( \sqrt{x^2} = |x| \), and:
\[
\sqrt{512} = \sqrt{256 \cdot 2} = 16\sqrt{2}
\]
So:
\[
\sqrt{512x^2} = 16\sqrt{2} \cdot |x| = 16|x|\sqrt{2}
\]
Then:
\[
8\sqrt{512x^2} = 8 \cdot 16|x|\sqrt{2} = 128|x|\sqrt{2}
\]

### Step 2: Combine the terms
Now, let's combine the two simplified expressions:
\[
8|x|\sqrt{2} + 128|x|\sqrt{2} = (8 + 128)|x|\sqrt{2} = 136|x|\sqrt{2}
\]

### Final Answer:
\[
2\sqrt{32x^2} + 8\sqrt{512x^2} = 136|x|\sqrt{2}
\]

Would you like further details or have any questions? 

Here are some related questions:
1. How do you simplify expressions involving square roots?
2. What are the properties of square roots used in algebra?
3. Can square roots be negative, and how does that affect simplification?
4. How does the absolute value function come into play when simplifying square roots of squared variables?
5. How can you combine like terms when dealing with radicals?

**Tip:** Always factor out perfect squares from square roots for easier simplification.

Learn how to simplify the expression 2√32x^2 + 8√512x^2 step-by-step. This problem involves key algebraic concepts such as radicals and the square root property, and is 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