To transform the expression \(\left(\sqrt[5]{x^7}\right)^3\) into an expression with a rational exponent, we should follow a step-by-step approach that involves understanding and applying the properties of exponents and radicals.
Firstly, recognize that the fifth root of \(x^7\) can be expressed as \(x^{7/5}\). This stems from the general property that the \(n\)-th root of \(x^m\) can be written as \(x^{m/n}\). Thus,
[tex]\[
\sqrt[5]{x^7} = x^{7/5}.
\][/tex]
Next, we need to raise this result to the power of 3. According to the rules of exponents, specifically the power of a power rule, \(\left( x^a \right)^b = x^{a \cdot b}\). Therefore,
[tex]\[
\left( x^{7/5} \right)^3 = x^{(7/5) \cdot 3}.
\][/tex]
Now, compute the product of the exponents:
[tex]\[
(7/5) \cdot 3 = 21/5.
\][/tex]
Putting it all together, the expression \(\left(\sqrt[5]{x^7}\right)^3\) simplifies to \(x^{21/5}\).
So, the transformation of [tex]\(\left(\sqrt[5]{x^7}\right)^3\)[/tex] results in [tex]\(x^{21/5}\)[/tex] expressed with a rational exponent.
