Use the properties of exponents to generate an equivalent expression for 3√8/√2

Question

Find the cube root of 8. Then, rewrite rational exponents as radicals and simplify.

Answer 1

To simplify the expression \( \frac{\sqrt[3]{8}}{\sqrt{2}} \), let's go through the steps as mentioned:

1. Find the cube root of 8:
   The cube root of \( 8 \) is \( 2 \), since \( 2^3 = 8 \). Therefore, we have: \[ \sqrt[3]{8} = 2 \]

2. Rewrite the expression:
   Now we can substitute this value back into the expression: \[ \frac{\sqrt[3]{8}}{\sqrt{2}} = \frac{2}{\sqrt{2}} \]

3. Simplify \(\frac{2}{\sqrt{2}}\):
   To simplify \( \frac{2}{\sqrt{2}} \), we can multiply the numerator and the denominator by \( \sqrt{2} \): \[ \frac{2}{\sqrt{2}} \cdot \frac{\sqrt{2}}{\sqrt{2}} = \frac{2\sqrt{2}}{2} \] This simplifies to: \[ \sqrt{2} \]

So, the equivalent expression for \( \frac{\sqrt[3]{8}}{\sqrt{2}} \) is \( \sqrt{2} \).