Question

Use properties of exponents to generate expression for ^3 square root 8/ square root 2

Answer 1

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:

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

2. 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:

\[
\sqrt{2}.
\]