Answer :
To represent [tex]\(\sqrt[6]{x^5}\)[/tex] in exponential form, we can start by understanding how roots can be converted into exponents.
The [tex]\(n\)[/tex]-th root of [tex]\(x\)[/tex], denoted as [tex]\(\sqrt[n]{x}\)[/tex], is equivalent to [tex]\(x\)[/tex] raised to the power of [tex]\(\frac{1}{n}\)[/tex]:
[tex]\[ \sqrt[n]{x} = x^{\frac{1}{n}} \][/tex]
Therefore, [tex]\(\sqrt[6]{x^5}\)[/tex] can be expressed where the exponent [tex]\(5\)[/tex] is divided by the root factor [tex]\(6\)[/tex]. This leads us to:
[tex]\[ \sqrt[6]{x^5} = \left(x^5\right)^{\frac{1}{6}} \][/tex]
When we raise a power to another power, we multiply the exponents:
[tex]\[ \left(x^5\right)^{\frac{1}{6}} = x^{5 \cdot \frac{1}{6}} = x^{\frac{5}{6}} \][/tex]
So, [tex]\(\sqrt[6]{x^5}\)[/tex] in exponential form is [tex]\(x^{\frac{5}{6}}\)[/tex].
Thus, the correct answer from the given options is:
[tex]\[ x^{\frac{5}{6}} \][/tex]
The [tex]\(n\)[/tex]-th root of [tex]\(x\)[/tex], denoted as [tex]\(\sqrt[n]{x}\)[/tex], is equivalent to [tex]\(x\)[/tex] raised to the power of [tex]\(\frac{1}{n}\)[/tex]:
[tex]\[ \sqrt[n]{x} = x^{\frac{1}{n}} \][/tex]
Therefore, [tex]\(\sqrt[6]{x^5}\)[/tex] can be expressed where the exponent [tex]\(5\)[/tex] is divided by the root factor [tex]\(6\)[/tex]. This leads us to:
[tex]\[ \sqrt[6]{x^5} = \left(x^5\right)^{\frac{1}{6}} \][/tex]
When we raise a power to another power, we multiply the exponents:
[tex]\[ \left(x^5\right)^{\frac{1}{6}} = x^{5 \cdot \frac{1}{6}} = x^{\frac{5}{6}} \][/tex]
So, [tex]\(\sqrt[6]{x^5}\)[/tex] in exponential form is [tex]\(x^{\frac{5}{6}}\)[/tex].
Thus, the correct answer from the given options is:
[tex]\[ x^{\frac{5}{6}} \][/tex]
