Math Problem Statement

2rad32x^2 + 8rad512x^2

Solution

Let's simplify the expression 232x2+8512x22\sqrt{32x^2} + 8\sqrt{512x^2}.

Step 1: Simplify the square roots

First, let's simplify each square root individually.

Simplifying 32x2\sqrt{32x^2}:

32x2=32x2\sqrt{32x^2} = \sqrt{32} \cdot \sqrt{x^2} Since x2=x\sqrt{x^2} = |x|, we have: 32=162=42\sqrt{32} = \sqrt{16 \cdot 2} = 4\sqrt{2} Thus: 32x2=42x=4x2\sqrt{32x^2} = 4\sqrt{2} \cdot |x| = 4|x|\sqrt{2} So: 232x2=24x2=8x22\sqrt{32x^2} = 2 \cdot 4|x|\sqrt{2} = 8|x|\sqrt{2}

Simplifying 512x2\sqrt{512x^2}:

512x2=512x2\sqrt{512x^2} = \sqrt{512} \cdot \sqrt{x^2} Again, x2=x\sqrt{x^2} = |x|, and: 512=2562=162\sqrt{512} = \sqrt{256 \cdot 2} = 16\sqrt{2} So: 512x2=162x=16x2\sqrt{512x^2} = 16\sqrt{2} \cdot |x| = 16|x|\sqrt{2} Then: 8512x2=816x2=128x28\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: 8x2+128x2=(8+128)x2=136x28|x|\sqrt{2} + 128|x|\sqrt{2} = (8 + 128)|x|\sqrt{2} = 136|x|\sqrt{2}

Final Answer:

232x2+8512x2=136x22\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.

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Math Problem Analysis

Mathematical Concepts

Algebra
Radicals
Simplifying Expressions

Formulas

√(a * b) = √a * √b
|x| = √(x^2)

Theorems

Square Root Property
Absolute Value Theorem

Suitable Grade Level

Grades 9-10