To simplify the expression [tex]\(\left(\frac{x^{1 / 2}}{x^{1 / 3}}\right)^{1 / 4}\)[/tex] to a single power of [tex]\(x\)[/tex], we can follow these steps:
1. Simplify the expression inside the parentheses first. We use the property of exponents that states [tex]\(\frac{a^m}{a^n} = a^{m-n}\)[/tex].
[tex]\[\frac{x^{1 / 2}}{x^{1 / 3}} = x^{(1 / 2) - (1 / 3)}\][/tex]
2. Next, we need to subtract the exponents [tex]\(\frac{1}{2}\)[/tex] and [tex]\(\frac{1}{3}\)[/tex]. In order to do this, we convert both fractions to have a common denominator. The least common denominator for 2 and 3 is 6.
[tex]\[
\frac{1}{2} = \frac{3}{6} \quad \text{and} \quad \frac{1}{3} = \frac{2}{6}
\][/tex]
3. Now we subtract the fractions:
[tex]\[
\frac{3}{6} - \frac{2}{6} = \frac{1}{6}
\][/tex]
So, [tex]\(\frac{x^{1 / 2}}{x^{1 / 3}}\)[/tex] simplifies to [tex]\(x^{1 / 6}\)[/tex].
4. Now consider the outer exponent, [tex]\((\cdot)^{1 / 4}\)[/tex]. We apply the rule [tex]\((a^m)^n = a^{m \cdot n}\)[/tex]:
[tex]\[
\left(x^{1 / 6}\right)^{1 / 4} = x^{(1 / 6) \cdot (1 / 4)}
\][/tex]
5. Multiply the exponents:
[tex]\[
\frac{1}{6} \cdot \frac{1}{4} = \frac{1}{24}
\][/tex]
6. Therefore, the simplified expression is:
[tex]\[
x^{1 / 24}
\][/tex]
Thus, [tex]\(\left(\frac{x^{1 / 2}}{x^{1 / 3}}\right)^{1 / 4}\)[/tex] simplifies to [tex]\(x^{1 / 24}\)[/tex].
