To simplify the expression \(\frac{\sqrt[3]{8}}{\sqrt{2}}\) using properties of exponents, we can start by rewriting the roots in terms of exponents:
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The cube root of 8 can be expressed as: \[ \sqrt[3]{8} = 8^{\frac{1}{3}}. \] Since \(8\) can be written as \(2^3\), we have: \[ 8^{\frac{1}{3}} = (2^3)^{\frac{1}{3}} = 2^{3 \cdot \frac{1}{3}} = 2^1 = 2. \]
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The square root of 2 can be expressed as: \[ \sqrt{2} = 2^{\frac{1}{2}}. \]
Now we can substitute these back into the expression:
\[ \frac{\sqrt[3]{8}}{\sqrt{2}} = \frac{2}{2^{\frac{1}{2}}}. \]
Next, we can apply the property of exponents that states \( \frac{a^m}{a^n} = a^{m-n} \):
\[ \frac{2}{2^{\frac{1}{2}}} = 2^{1 - \frac{1}{2}} = 2^{\frac{1}{2}}. \]
Finally, we can express \(2^{\frac{1}{2}}\) as the square root:
\[ 2^{\frac{1}{2}} = \sqrt{2}. \]
Thus, the simplified expression for \(\frac{\sqrt[3]{8}}{\sqrt{2}}\) is:
\[ \sqrt{2}. \]