D. Rationalizing the Denominator: Monomial Denominators
Completion requirements
D. Rationalizing the Denominator: Monomial Denominators
Investigation |
Removing Radicals
Simplified radical expressions do not have radicals in the denominator. The process of removing the radicals is called rationalizing the denominator. To rationalize the denominator, you will apply the rule \(\sqrt[n]{{x^n }} = x\).
Denominators with Square Roots
For rational expressions with square root denominators, multiply the denominator by itself in order to remove the radical sign. You must also remember to multiply the numerator by the same value so that you do not change the expression! This is similar to saying you have multiplied by \(1\), since \(\frac{{\sqrt a }}{{\sqrt a }} = 1\). This process is demonstrated in Example 1.
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Expand and simplify the following expressions.
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\(\left( {3\sqrt {13} } \right)\left( {\sqrt {13} } \right)\)
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\(\left( {\sqrt[3]{2}} \right)\left( {\sqrt[3]{2}} \right)^2 \)
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What do you notice about each of the answers in the Investigation?
Simplified radical expressions do not have radicals in the denominator. The process of removing the radicals is called rationalizing the denominator. To rationalize the denominator, you will apply the rule \(\sqrt[n]{{x^n }} = x\).
Denominators with Square Roots
For rational expressions with square root denominators, multiply the denominator by itself in order to remove the radical sign. You must also remember to multiply the numerator by the same value so that you do not change the expression! This is similar to saying you have multiplied by \(1\), since \(\frac{{\sqrt a }}{{\sqrt a }} = 1\). This process is demonstrated in Example 1.