use the properties of exponents to generate an equivalent expression for (\root(3)(8))/(\sqrt(2))

2^

1 answer

To simplify the expression \(\frac{\sqrt[3]{8}}{\sqrt{2}}\) using the properties of exponents, we start by rewriting each component in terms of base \(2\).

  1. Rewrite \(\sqrt[3]{8}\): \[ 8 = 2^3 \implies \sqrt[3]{8} = (2^3)^{\frac{1}{3}} = 2^{3 \cdot \frac{1}{3}} = 2^1 = 2. \]

  2. Rewrite \(\sqrt{2}\): \[ \sqrt{2} = 2^{\frac{1}{2}}. \]

Now we substitute these rewritten expressions into the original expression: \[ \frac{\sqrt[3]{8}}{\sqrt{2}} = \frac{2}{\sqrt{2}} = \frac{2}{2^{\frac{1}{2}}}. \]

  1. Use the properties of exponents: Recall the property of exponents \(\frac{a^m}{a^n} = a^{m-n}\). So we have: \[ \frac{2}{2^{\frac{1}{2}}} = 2^{1 - \frac{1}{2}} = 2^{\frac{1}{2}}. \]

Thus, the equivalent expression is: \[ 2^{\frac{1}{2}}. \]

Finally, we can summarize our finding: \[ \frac{\sqrt[3]{8}}{\sqrt{2}} = 2^{\frac{1}{2}}. \]