Sure! Let's solve the expression [tex]\(\frac{\sqrt[3]{x^7}}{x^2}\)[/tex].
1. Start with simplifying the numerator [tex]\(\sqrt[3]{x^7}\)[/tex]. The cube root of [tex]\(x^7\)[/tex] can be written using rational exponents:
[tex]\[
\sqrt[3]{x^7} = x^{7/3}
\][/tex]
2. Now, we substitute this into the original expression:
[tex]\[
\frac{x^{7/3}}{x^2}
\][/tex]
3. Next, we apply the properties of exponents for division. Recall that [tex]\(\frac{a^m}{a^n} = a^{m-n}\)[/tex]:
[tex]\[
\frac{x^{7/3}}{x^2} = x^{7/3 - 2}
\][/tex]
4. To proceed, we need to subtract the exponents. Note that [tex]\(2\)[/tex] can be written as a fraction with the same denominator as [tex]\(\frac{7}{3}\)[/tex]:
[tex]\[
2 = \frac{6}{3}
\][/tex]
5. Now, subtract the fraction exponents:
[tex]\[
x^{7/3 - 6/3} = x^{(7-6)/3} = x^{1/3}
\][/tex]
So, the simplified form of the given expression [tex]\(\frac{\sqrt[3]{x^7}}{x^2}\)[/tex] is:
[tex]\[
x^{1/3}
\][/tex]
In numerical approximation form, this exponent [tex]\( \frac{1}{3} \)[/tex] is approximately [tex]\( 0.333333333333333 \)[/tex]. Hence, [tex]\( x^{1/3} \)[/tex] can also be written as:
[tex]\[
x^{0.333333333333333}
\][/tex]
