To find an equivalent expression for \(\frac{\sqrt[3]{8}}{\sqrt{2}}\), we will proceed through the following steps.
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Find the cube root of 8: \[ \sqrt[3]{8} = 2 \] This is because \(2^3 = 8\).
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Rewrite the expression: The expression \(\frac{\sqrt[3]{8}}{\sqrt{2}}\) can now be simplified as: \[ \frac{2}{\sqrt{2}} \]
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Rewrite \(\sqrt{2}\) as a rational exponent: Recall that \(\sqrt{2}\) can be expressed as: \[ \sqrt{2} = 2^{1/2} \] Therefore, the expression becomes: \[ \frac{2}{2^{1/2}} \]
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Use the properties of exponents to simplify: We can rewrite \(2\) as \(2^{1}\) so that: \[ \frac{2^{1}}{2^{1/2}} = 2^{1 - 1/2} = 2^{1/2} \]
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Rewrite \(2^{1/2}\) as a radical: Thus: \[ 2^{1/2} = \sqrt{2} \]
So the final equivalent expression for \(\frac{\sqrt[3]{8}}{\sqrt{2}}\) is: \[ \sqrt{2} \]
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