Rationalize each denominator. See Example 8. 12 —— √72

Verified step by step guidance

1. Identify the denominator that needs to be rationalized: \( \sqrt{72} \).
2. Express \( \sqrt{72} \) in terms of its prime factors: \( \sqrt{72} = \sqrt{36 \times 2} = \sqrt{36} \times \sqrt{2} = 6\sqrt{2} \).
3. Multiply both the numerator and the denominator by \( \sqrt{2} \) to rationalize the denominator: \( \frac{12}{6\sqrt{2}} \times \frac{\sqrt{2}}{\sqrt{2}} \).
4. Simplify the expression: The denominator becomes \( 6 \times 2 = 12 \) and the numerator becomes \( 12\sqrt{2} \).
5. Simplify the fraction: \( \frac{12\sqrt{2}}{12} = \sqrt{2} \).

Key Concepts

Rationalizing the Denominator

Rationalizing the denominator involves rewriting a fraction so that the denominator is a rational number. This is often done by multiplying both the numerator and the denominator by a suitable expression that eliminates any irrational numbers, such as square roots. The goal is to simplify the expression while maintaining its value.

Square Roots

A square root of a number is a value that, when multiplied by itself, gives the original number. For example, the square root of 72 can be simplified to 6√2, since 72 = 36 × 2 and √36 = 6. Understanding how to simplify square roots is essential for rationalizing denominators effectively.

Multiplying by Conjugates

When dealing with expressions that contain square roots, multiplying by the conjugate can be a useful technique. The conjugate of a binomial expression is formed by changing the sign between two terms. This method is particularly helpful when the denominator is a binomial involving square roots, as it can eliminate the irrational part when multiplied.