Rationalize each denominator. See Example 8. 4 —— √6

Verified step by step guidance

1. Identify the expression to rationalize: \(\frac{4}{\sqrt{6}}\).
2. Recall that to rationalize a denominator containing a square root, multiply both numerator and denominator by the same square root to eliminate the radical in the denominator.
3. Multiply numerator and denominator by \(\sqrt{6}\): \(\frac{4}{\sqrt{6}} \times \frac{\sqrt{6}}{\sqrt{6}}\).
4. Simplify the numerator: \(4 \times \sqrt{6} = 4\sqrt{6}\).
5. Simplify the denominator using the property \(\sqrt{a} \times \sqrt{a} = a\): \(\sqrt{6} \times \sqrt{6} = 6\), so the denominator becomes 6.

Key Concepts

Rationalizing the Denominator

Rationalizing the denominator involves eliminating any square roots or irrational numbers from the denominator of a fraction. This is done by multiplying both numerator and denominator by a suitable radical that will make the denominator a rational number, typically by using the conjugate or the same radical.

Properties of Square Roots

Square roots follow specific properties such as √a × √a = a and √a / √b = √(a/b). Understanding these properties helps simplify expressions and perform operations like rationalization by converting radicals into whole numbers or simpler radicals.

Multiplying Fractions by 1

Multiplying a fraction by a form of 1, such as √6/√6, changes the expression without altering its value. This technique is essential in rationalizing denominators because it allows the denominator to be transformed into a rational number while keeping the fraction equivalent.