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

Verified step by step guidance

1. Start by simplifying the denominator \( \sqrt{72} \). Factor 72 into its prime factors: \( 72 = 36 \times 2 \), so \( \sqrt{72} = \sqrt{36 \times 2} \).
2. Use the property of square roots that \( \sqrt{a \times b} = \sqrt{a} \times \sqrt{b} \) to rewrite \( \sqrt{72} \) as \( \sqrt{36} \times \sqrt{2} \).
3. Since \( \sqrt{36} = 6 \), simplify the denominator to \( 6 \sqrt{2} \). Now the expression is \( \frac{12}{6 \sqrt{2}} \).
4. To rationalize the denominator, multiply both numerator and denominator by \( \sqrt{2} \) to eliminate the square root from the denominator: \( \frac{12}{6 \sqrt{2}} \times \frac{\sqrt{2}}{\sqrt{2}} \).
5. Multiply the numerators and denominators separately: numerator becomes \( 12 \times \sqrt{2} \), denominator becomes \( 6 \times \sqrt{2} \times \sqrt{2} = 6 \times 2 \). Simplify the resulting expression.

Key Concepts

Simplifying Radicals

Simplifying radicals involves expressing the radicand as a product of perfect squares and other factors to rewrite the square root in simplest form. For example, √72 can be broken down into √(36 × 2), which simplifies to 6√2. This step makes further operations like rationalization easier.

Rationalizing the Denominator

Rationalizing the denominator means eliminating any radicals from the denominator of a fraction by multiplying numerator and denominator by an appropriate radical. This process converts the denominator into a rational number, making the expression simpler and more standard.

Properties of Square Roots

The properties of square roots, such as √a × √b = √(a × b) and (√a)² = a, are essential for manipulating and simplifying expressions involving radicals. These properties allow for breaking down and combining radicals during simplification and rationalization.