Question

Express (6^3)^1/2 in simplest form

Answer 1

To simplify \((6^3)^{1/2}\), you can use the property of exponents that states \((a^m)^n = a^{m \cdot n}\).

So, we can rewrite the expression as follows:

\[
(6^3)^{1/2} = 6^{3 \cdot \frac{1}{2}} = 6^{3/2}
\]

Next, to express \(6^{3/2}\) in simplest form, you can separate the exponent:

\[
6^{3/2} = (6^3)^{1/2} = \sqrt{6^3} = \sqrt{216}
\]

Calculating \(\sqrt{216}\):

\[
\sqrt{216} = \sqrt{36 \times 6} = \sqrt{36} \times \sqrt{6} = 6\sqrt{6}
\]

Alternatively, you can also express \(6^{3/2}\) in fractional form:

\[
6^{3/2} = \frac{6^3}{6^1} = \frac{216}{6} = 36
\]

But the simplest radical form of \((6^3)^{1/2}\) is \(6\sqrt{6}\).

Both forms \(6^{3/2}\) and \(6\sqrt{6}\) are valid, so the final answer can be:

\[
6^{3/2} = 6\sqrt{6}
\] or simply \(6^{3/2}\).