Simplify the expression to a single power of [tex] x [/tex]:

[tex] \left(\frac{x^{1 / 2}}{x^{1 / 3}}\right)^{1 / 4} [/tex]



Answer :

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].