Questions: (√[4]16 x)^3 use rational exponents to put this in different form

Solution Steps

To express \((\sqrt[4]{16x})^3\) using rational exponents, we first convert the fourth root into a rational exponent. The fourth root of a number is the same as raising that number to the power of \(1/4\). Therefore, \(\sqrt[4]{16x}\) can be written as \((16x)^{1/4}\). Raising this expression to the power of 3 involves multiplying the exponents.

The expression given is \((\sqrt[4]{16x})^3\). The fourth root of a number can be expressed using rational exponents as raising the number to the power of \(\frac{1}{4}\). Therefore, we can rewrite \(\sqrt[4]{16x}\) as:

\[ (16x)^{\frac{1}{4}} \]

The expression \((\sqrt[4]{16x})^3\) can now be rewritten using the expression from Step 1:

\[ ((16x)^{\frac{1}{4}})^3 \]

According to the power of a power rule, \((a^m)^n = a^{m \cdot n}\). Applying this rule, we have:

\[ (16x)^{\frac{1}{4} \cdot 3} = (16x)^{\frac{3}{4}} \]

Final Answer