To simplify the expression \( \frac{3\sqrt{8}}{\sqrt{2}} \) using the properties of exponents, we can start by rewriting the square roots in terms of exponents:
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Rewriting the square roots: \[ \sqrt{8} = 8^{1/2}, \quad \text{and} \quad \sqrt{2} = 2^{1/2} \]
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Rewriting \( 8 \) in terms of powers of \( 2 \): \[ 8 = 2^3 \] Therefore, \[ \sqrt{8} = (2^3)^{1/2} = 2^{3/2} \]
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Substitute back into the original expression: \[ \frac{3\sqrt{8}}{\sqrt{2}} = \frac{3 \cdot 2^{3/2}}{2^{1/2}} \]
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Using the property of exponents \( \frac{a^m}{a^n} = a^{m-n} \): \[ \frac{2^{3/2}}{2^{1/2}} = 2^{3/2 - 1/2} = 2^{2/2} = 2^1 = 2 \]
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Now, substitute this back: \[ 3 \cdot \frac{2^{3/2}}{2^{1/2}} = 3 \cdot 2 = 6 \]
Thus, the equivalent expression for \( \frac{3\sqrt{8}}{\sqrt{2}} \) is: \[ 6 \]
So the final expression is \( 6 \).
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