Questions: Express (8^3)^(1/5) in simplest radical form.

Solution Steps

To express \((8^{3})^{\frac{1}{5}}\) in simplest radical form, we can use the property of exponents that states \((a^m)^n = a^{m \cdot n}\). First, calculate the exponent \(3 \cdot \frac{1}{5}\), which simplifies the expression to \(8^{\frac{3}{5}}\). Then, express this in radical form as \(\sqrt[5]{8^3}\).

We start with the expression \((8^{3})^{\frac{1}{5}}\). Using the property of exponents, we can rewrite this as: \[ 8^{\frac{3}{5}} \]

Next, we express \(8^{\frac{3}{5}}\) in radical form: \[ 8^{\frac{3}{5}} = \sqrt[5]{8^3} \]

Calculating \(8^3\): \[ 8^3 = 512 \] Thus, we have: \[ \sqrt[5]{512} \]

The number \(512\) can be expressed as \(2^9\) since \(512 = 2^9\). Therefore: \[ \sqrt[5]{512} = \sqrt[5]{2^9} = 2^{\frac{9}{5}} = 2^{1 + \frac{4}{5}} = 2 \cdot 2^{\frac{4}{5}} \]

Final Answer