Answered

Which expression is equivalent to $\sqrt[6]{g^5}$, if $g>0$ ?


A$g^{\frac{6}{5}}$
B$5 g^6$
C$\frac{5}{6} g$
D$g^{\frac{5}{6}}$

Verified Answer :

  • Recognize the radical expression \(\sqrt[6]{g^5}\) where \(g>0\).
  • Apply the radical-to-exponent conversion rule \(\sqrt[n]{x^m} = x^{\frac{m}{n}}\).
  • Identify that \(n=6\) and \(m=5\), which leads to the equivalent exponent \(\frac{5}{6}\).
  • Match the result \(g^{\frac{5}{6}}\) with the correct option.
  • The equivalent expression is \(\boxed{g^{\frac{5}{6}}}\).

Explanation

  1. Analyzing the Given Expression

We are given the expression \(\sqrt[6]{g^5}\) where \(g > 0\). Our goal is to rewrite this radical expression in its equivalent exponential form. Radicals are another way of expressing fractional exponents. The fundamental property relating radicals and exponents is \(\sqrt[n]{x^m} = x^{\frac{m}{n}}\), where \(x\) is the base, \(n\) is the index of the root, and \(m\) is the power to which the base is raised.

  1. Converting to Exponential Form

In the expression \(\sqrt[6]{g^5}\), we can identify the following components:

  • The base \(x\) is \(g\).
  • The index of the root \(n\) is \(6\).
  • The power \(m\) to which the base is raised is \(5\).

Using the general rule \(\sqrt[n]{x^m} = x^{\frac{m}{n}}\), we substitute these identified values into the formula to obtain \(g^{\frac{5}{6}}\).

  1. Conclusion and Selection

Comparing our result, \(g^{\frac{5}{6}}\), with the provided options:

  • \(g^{\frac{6}{5}}\)
  • \(5 g^6\)
  • \(\frac{5}{6} g\)
  • \(g^{\frac{5}{6}}\)

The correct expression is the fourth one listed. Thus, \(\sqrt[6]{g^5} = \boxed{g^{\frac{5}{6}}}\).

Examples

In many fields of science, such as chemistry or physics, measurements are often expressed in radical form for convenience, but they must be converted to exponential form to be plugged into complex formulas or computer algorithms. For instance, if you are modeling the growth of a substance over time using the relationship \(V = \sqrt[6]{g^5}\), converting it to \(V = g^{5/6}\) makes it significantly easier to perform calculations like finding the derivative of growth with respect to \(g\), which represents the rate of change. Mastering this conversion is essential for simplifying algebraic equations and performing calculus operations accurately.

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