Math Problem Statement

Simplify the given expression: √(10x³) * √(5x²)

Solution

Let's simplify the given expression, which is:

10x35x2\sqrt{10x^3} \cdot \sqrt{5x^2}

We can first combine the square roots using the property:

ab=ab\sqrt{a} \cdot \sqrt{b} = \sqrt{a \cdot b}

So we get:

10x35x2=50x5\sqrt{10x^3 \cdot 5x^2} = \sqrt{50x^5}

Next, simplify the square root:

50x5=252x4x=25x42x\sqrt{50x^5} = \sqrt{25 \cdot 2 \cdot x^4 \cdot x} = \sqrt{25x^4} \cdot \sqrt{2x}

Since 25x4=5x2\sqrt{25x^4} = 5x^2, the expression becomes:

5x22x5x^2 \cdot \sqrt{2x}

Thus, the simplified form is:

5x22x5x^2 \sqrt{2x}

This matches the last answer choice.

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Related Questions:

  1. How do you simplify products of square roots in general?
  2. What are the properties of exponents and how are they used in simplifications?
  3. What is the general rule for taking square roots of variables with exponents?
  4. How do you factor square roots to simplify expressions?
  5. How would the process change if there were cube roots instead of square roots?

Tip:

When dealing with square roots, always try to simplify the radicand (the number inside the square root) by factoring out perfect squares.

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Math Problem Analysis

Mathematical Concepts

Simplifying square roots
Multiplication of square roots
Exponents

Formulas

√a * √b = √(a * b)
√(x^n) = x^(n/2) when n is even

Theorems

Square root property: √(a) * √(b) = √(a * b)

Suitable Grade Level

Grades 9-11