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Rewrite the rational exponent as a radical by extending the properties of integer exponents. 2 to the 3 over 4 power, all over 2 to the 1 over 2 power the eighth root of 2 to the third power - THIS ONE the square root of 2 to the 3 over 4 power the fourth root of 2 the square root of 2

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MorganTyyler

The answer is the fourth root of 2. 2 to the 3 over 4 power is [latex]2^{ \frac{3}{4} } [/latex] 2 to the 1 over 2 power is [latex] 2^{ \frac{1}{2} } [/latex] 2 to the 3 over 4 power, all over 2 to the 1 over 2 power is [latex] \frac{2^{ \frac{3}{4} } }{2^{ \frac{1}{2} }} [/latex] So, use the rule: [latex] \frac{x^{a} }{ x^{b} } = x^{a-b} [/latex] [latex]\frac{2^{ \frac{3}{4} } }{2^{ \frac{1}{2} }} = 2^{\frac{3}{4}- \frac{1}{2}}= 2^{\frac{3}{4}- \frac{1*2}{2*2}}= 2^{\frac{3}{4}- \frac{2}{4}}= 2^{ \frac{3-2}{4} } = 2^{ \frac{1}{4} } [/latex] Now, use the rule: [latex] a^{ \frac{m}{n}} = \sqrt[n]{ x^{m} } [/latex] [latex]2^{ \frac{1}{4} } = \sqrt[4]{ 2^{1} }= \sqrt[4]{2} [/latex] which is the same as the fourth root of 2.

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