To simplify the expression \(\frac{\sqrt[3]{8}}{\sqrt{2}}\), we can use properties of exponents.
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First, rewrite the roots in exponential form: \[ \sqrt[3]{8} = 8^{1/3} \quad \text{and} \quad \sqrt{2} = 2^{1/2} \]
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Next, express \(8\) as a power of \(2\): \[ 8 = 2^3 \] Thus, \[ \sqrt[3]{8} = (2^3)^{1/3} = 2^{3 \cdot \frac{1}{3}} = 2^1 = 2 \]
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Now substitute \(2\) back into the original expression: \[ \frac{\sqrt[3]{8}}{\sqrt{2}} = \frac{2}{2^{1/2}} \]
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Using the properties of exponents: \[ \frac{2}{2^{1/2}} = 2^{1 - \frac{1}{2}} = 2^{\frac{1}{2}} \]
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Finally, rewrite \(2^{1/2}\) in radical form: \[ 2^{\frac{1}{2}} = \sqrt{2} \]
Putting it all together, the equivalent expression for \(\frac{\sqrt[3]{8}}{\sqrt{2}}\) is: \[ \sqrt{2} \]