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Laws of Radicals

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Rationalizing the Denominator

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To get rid of radicals in the denominator, multiply numerator and denominator by a radical that will give the denominator a perfect square under the radical

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Roots of Fractional Exponents

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an=a1n\sqrt[n]{a} = a^{\frac{1}{n}}, where aa is a nonnegative real number

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Product Property of Radicals

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ab=ab\sqrt{a} \cdot \sqrt{b} = \sqrt{ab}, where aa and bb are nonnegative

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Simplifying Radicals with Perfect Powers

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If a=bna = b^n for some integer nn, then an=b\sqrt[n]{a} = b provided that bb and aa are nonnegative

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Quotient Property of Radicals

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ab=ab\frac{\sqrt{a}}{\sqrt{b}} = \sqrt{\frac{a}{b}}, where a0a \ge 0 and b>0b > 0

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Laws of Exponents for Radicals

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When radicals have the same index, laws of exponents like aman=am+na^m \cdot a^n = a^{m+n} apply to their radicands

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Simplifying Nested Radicals

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a=a4\sqrt{\sqrt{a}} = \sqrt[4]{a}, where aa is a nonnegative real number

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Power Property of Radicals

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(am)n=anm(\sqrt[m]{a})^n = \sqrt[m]{a^n}, where aa is a nonnegative real number

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