Math Problem Statement

Rationalize the denominator for the expression 4x^2 / √(5x).

Solution

To rationalize the denominator of the given expression:

4x25x\frac{4x^2}{\sqrt{5x}}

we need to eliminate the square root from the denominator. This is typically done by multiplying both the numerator and the denominator by the conjugate or an appropriate factor that will clear the square root.

Steps to Rationalize:

  1. Identify the denominator: The denominator is 5x\sqrt{5x}.
  2. Multiply by the appropriate factor: To eliminate the square root in the denominator, we multiply both the numerator and the denominator by 5x\sqrt{5x}.

4x25x×5x5x=4x25x5x\frac{4x^2}{\sqrt{5x}} \times \frac{\sqrt{5x}}{\sqrt{5x}} = \frac{4x^2 \cdot \sqrt{5x}}{5x}

  1. Simplify the expression:

    • The numerator becomes 4x25x4x^2 \sqrt{5x}.
    • The denominator simplifies to 5x5x.

Thus, the rationalized form is:

4x25x5x\frac{4x^2 \sqrt{5x}}{5x}

  1. Simplify further if possible:

Since there is an xx in both the numerator and the denominator, we can cancel one xx:

4x5x5\frac{4x \sqrt{5x}}{5}

Final Answer:

The expression with a rationalized denominator is:

4x5x5\frac{4x \sqrt{5x}}{5}

Would you like further details, or do you have any questions?

Related Questions:

  1. How do you rationalize the denominator if it contains a binomial with a square root?
  2. What is the general method for simplifying square roots in the denominator?
  3. How do you multiply and divide expressions with radicals?
  4. Can you explain how to simplify higher-order roots in a denominator?
  5. What happens when you have a square root in both the numerator and the denominator?

Tip:

When rationalizing denominators with radicals, always look for the simplest way to remove the radical by multiplying by its conjugate or an appropriate factor.

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Math Problem Analysis

Mathematical Concepts

Algebra
Rationalizing the Denominator
Radicals

Formulas

Multiplication of Radicals: √a * √a = a
Simplification of Fractions: (a * b) / c = ab / c

Theorems

Properties of Radicals
Simplification of Expressions

Suitable Grade Level

Grades 9-10