Question

express in simplest radical form.
\\((3125x^{3})^{\frac{6}{5}}\\)

Explanation:

Step1: Recognize the exponent rule

We use the power of a product rule \((ab)^n = a^n b^n\) and the power of a power rule \((a^m)^n=a^{mn}\). First, apply the power of a product rule to \((3125x^{3})^{\frac{6}{5}}\), we get \(3125^{\frac{6}{5}}\times(x^{3})^{\frac{6}{5}}\).

Step2: Simplify \(3125^{\frac{6}{5}}\)

We know that \(3125 = 5^{5}\), so \(3125^{\frac{6}{5}}=(5^{5})^{\frac{6}{5}}\). Using the power of a power rule \((a^m)^n = a^{mn}\), we have \(5^{5\times\frac{6}{5}}=5^{6}\). Calculate \(5^{6}=15625\).

Step3: Simplify \((x^{3})^{\frac{6}{5}}\)

Using the power of a power rule \((a^m)^n=a^{mn}\), we get \(x^{3\times\frac{6}{5}} = x^{\frac{18}{5}}\). We can rewrite \(x^{\frac{18}{5}}\) as \(x^{3+\frac{3}{5}}=x^{3}\cdot x^{\frac{3}{5}}\) or in radical form \(x^{3}\sqrt[5]{x^{3}}\). But we can also keep it as \(x^{\frac{18}{5}}\) or rewrite the whole expression. Wait, actually, let's re - evaluate the simplification of \(3125^{\frac{6}{5}}\) again. Wait, \(3125 = 5^{5}\), so \((5^{5})^{\frac{6}{5}}=5^{5\times\frac{6}{5}} = 5^{6}\)? Wait, no, \(5^{5\times\frac{6}{5}}=5^{6}\)? Wait, \(5\times\frac{6}{5}=6\), yes. But \(5^{6}=15625\). And \((x^{3})^{\frac{6}{5}}=x^{\frac{18}{5}}=x^{3+\frac{3}{5}}=x^{3}\cdot x^{\frac{3}{5}}\) or \(x^{3}\sqrt[5]{x^{3}}\). But let's do it step by step correctly.

Wait, another way: \((ab)^n=a^n b^n\), so \((3125x^{3})^{\frac{6}{5}}=3125^{\frac{6}{5}}\times x^{3\times\frac{6}{5}}\)

We know that \(3125 = 5^{5}\), so \(3125^{\frac{6}{5}}=(5^{5})^{\frac{6}{5}}=5^{5\times\frac{6}{5}} = 5^{6}=15625\)

And \(x^{3\times\frac{6}{5}}=x^{\frac{18}{5}}=x^{3+\frac{3}{5}}=x^{3}\cdot x^{\frac{3}{5}}=x^{3}\sqrt[5]{x^{3}}\)

But we can also write the entire expression as \(15625x^{\frac{18}{5}}\) or \(15625x^{3}\sqrt[5]{x^{3}}\). Wait, but let's check the exponent calculation again. Wait, \(3\times\frac{6}{5}=\frac{18}{5}=3\frac{3}{5}\), so \(x^{\frac{18}{5}}=x^{3}\cdot x^{\frac{3}{5}}\), and \(x^{\frac{3}{5}}=\sqrt[5]{x^{3}}\).

But let's re - express the original problem:

\((3125x^{3})^{\frac{6}{5}}=(5^{5}x^{3})^{\frac{6}{5}}\)

Using \((a\times b)^n=a^n\times b^n\), we have \((5^{5})^{\frac{6}{5}}\times(x^{3})^{\frac{6}{5}}\)

\((5^{5})^{\frac{6}{5}} = 5^{5\times\frac{6}{5}}=5^{6}=15625\)

\((x^{3})^{\frac{6}{5}}=x^{3\times\frac{6}{5}}=x^{\frac{18}{5}}\)

So the expression in simplest radical form is \(15625x^{\frac{18}{5}}\) or we can write \(x^{\frac{18}{5}}\) as \(x^{3}\sqrt[5]{x^{3}}\), so \(15625x^{3}\sqrt[5]{x^{3}}\)

Wait, but maybe there is a simpler way. Wait, \(3125^{\frac{6}{5}}\): Let's compute \(3125^{\frac{1}{5}}\) first, since the denominator of the fraction exponent is 5. \(3125^{\frac{1}{5}} = 5\) (because \(5^{5}=3125\)), then \(3125^{\frac{6}{5}}=(3125^{\frac{1}{5}})^{6}=5^{6}=15625\)

And \((x^{3})^{\frac{6}{5}}=x^{\frac{18}{5}}\), so the expression is \(15625x^{\frac{18}{5}}\) or \(15625x^{3}\sqrt[5]{x^{3}}\)

Answer:

\(15625x^{\frac{18}{5}}\) (or \(15625x^{3}\sqrt[5]{x^{3}}\))