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

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Identify the expression to rationalize: \(\frac{4}{\sqrt{6}}\).

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.

Multiply numerator and denominator by \(\sqrt{6}\): \(\frac{4}{\sqrt{6}} \times \frac{\sqrt{6}}{\sqrt{6}}\).

Simplify the numerator: \(4 \times \sqrt{6} = 4\sqrt{6}\).

Simplify the denominator using the property \(\sqrt{a} \times \sqrt{a} = a\): \(\sqrt{6} \times \sqrt{6} = 6\), so the denominator becomes 6.

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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.

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Rationalizing Denominators

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.

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Imaginary Roots with the Square Root Property

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.

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