Answered

Select the correct answer.

Which expression is equivalent to [tex]\sqrt[6]{g^5}[/tex], if [tex]g\ \textgreater \ 0[/tex]?

A. [tex]g^{\frac{5}{5}}[/tex]

B. [tex]5 g^6[/tex]

C. [tex]\frac{5}{6} g[/tex]

D. [tex]g^{\frac{5}{6}}[/tex]



Answer :

GinnyAnswer
To determine the equivalent expression for [tex]\(\sqrt[6]{g^5}\)[/tex], we need to convert the radical expression into an expression with exponents.

The general form for converting a [tex]\(n\)[/tex]th root of [tex]\(x^m\)[/tex] is given by:
[tex]\[ \sqrt[n]{x^m} = x^{\frac{m}{n}} \][/tex]

Applying this rule to [tex]\(\sqrt[6]{g^5}\)[/tex]:
[tex]\[ \sqrt[6]{g^5} = g^{\frac{5}{6}} \][/tex]

Now, let's compare this result with the given choices:

A. [tex]\(g^{\frac{5}{5}}\)[/tex] simplifies to [tex]\(g^1\)[/tex], which is just [tex]\(g\)[/tex].

B. [tex]\(5 g^6\)[/tex] is simply [tex]\(5 \times g^6\)[/tex], which is not in the form we need.

C. [tex]\(\frac{5}{6} g\)[/tex] is a linear term, not an exponential form that matches our converted expression.

D. [tex]\(g^{\frac{5}{6}}\)[/tex] directly matches our result from the conversion.

Therefore, the correct expression equivalent to [tex]\(\sqrt[6]{g^5}\)[/tex] is:
[tex]\[ \boxed{D} \][/tex]