Rationalize each denominator. See Example 8. 18 —— √27

Verified step by step guidance

1. Identify the expression to rationalize: \(\frac{18}{\sqrt{27}}\).
2. Simplify the square root in the denominator if possible. Since \(\sqrt{27} = \sqrt{9 \times 3} = \sqrt{9} \times \sqrt{3} = 3\sqrt{3}\), rewrite the expression as \(\frac{18}{3\sqrt{3}}\).
3. Simplify the fraction by dividing the numerator and denominator by 3: \(\frac{18}{3\sqrt{3}} = \frac{6}{\sqrt{3}}\).
4. To rationalize the denominator, multiply both numerator and denominator by \(\sqrt{3}\) to eliminate the square root in the denominator: \(\frac{6}{\sqrt{3}} \times \frac{\sqrt{3}}{\sqrt{3}}\).
5. Multiply the numerators and denominators: numerator becomes \(6 \times \sqrt{3}\), denominator becomes \(\sqrt{3} \times \sqrt{3} = 3\). The expression is now \(\frac{6\sqrt{3}}{3}\). You can simplify this further by dividing numerator and denominator by 3.

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, √27 can be simplified to 3√3 because 27 = 9 × 3 and √9 = 3.

Rationalizing the Denominator

Rationalizing the denominator means eliminating any radicals from the denominator of a fraction by multiplying numerator and denominator by a suitable radical expression. This process makes the denominator a rational number, which is often preferred in final answers.

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 the combination, separation, and simplification of square root terms.