rational exponents

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What is a rational exponent?

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@@ ext{The radical form is } \sqrt{x}.$$

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What is the relationship between rational exponents and radicals?

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@@x^{rac{3}{2} + rac{1}{2}} = x^{rac{4}{2}} = x^2.$$

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@@8^{rac{2}{3}} = \left(\sqrt[3]{8}\right)^2 = 2^2 = 4.$$

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@@\sqrt[4]{x^3}.$$

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@@x^{rac{5}{3} + rac{1}{3}} \cdot y^{rac{2}{3}} = x^{rac{6}{3}} \cdot y^{rac{2}{3}} = x^2 \cdot y^{rac{2}{3}}.$$

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@@x^{-rac{1}{2}}.$$

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@@x^{rac{3}{2} - rac{1}{2}} = x^{rac{2}{2}} = x.$$

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@@\sqrt{x^5} = x^{2} \cdot \sqrt{x}.$$