Verified Answer :

  • Recall the definition of fractional exponents: \(\sqrt[n]{x^m} = x^{\frac{m}{n}}\).
  • Identify the index of the radical \(n = 5\) and the power of the base \(m = 3\).
  • Substitute these into the formula to obtain \(x^{\frac{3}{5}}\).
  • The final exponential form is \(\boxed{x^{\frac{3}{5}}}\).

Explanation

  1. Understanding the Relationship between Radicals and Exponents

To represent a radical expression in exponential form, we rely on the fundamental relationship between roots and powers. The general rule for converting a radical to an exponent is:
\[\sqrt[n]{a^m} = a^{\frac{m}{n}}\]
where \(a\) is the base, \(m\) is the power of the radicand, and \(n\) is the index of the root. This conversion is a standard algebraic technique that allows us to treat radicals as fractional exponents, making calculations involving powers much simpler.

  1. Applying the Conversion Rule

In the given expression \(\sqrt[5]{x^3}\), we can identify the specific components needed to apply our rule:

  • The base \(a\) is \(x\).
  • The power of the radicand \(m\) is \(3\).
  • The index of the radical \(n\) is \(5\).

By substituting these values into our general formula, we get:
\[\sqrt[5]{x^3} = x^{\frac{3}{5}}\]
This effectively expresses the radical form in the required exponential form \(x^k\), where \(k = \frac{3}{5}\).

Examples

This conversion is incredibly useful in fields like engineering and science. For instance, if you are working with formulas for the volume of complex shapes or growth models in biology that involve root functions, converting them to exponential form makes them much easier to handle using basic exponent laws. For example, if you need to calculate the derivative or integral of \(\sqrt[5]{x^3}\), it is far easier to work with \(x^{3/5}\) than it is to work with the radical symbol itself. It simplifies your algebraic manipulations significantly.

Related Questions