Sure! Let's simplify the expression [tex]\(\frac{x^{\frac{5}{6}}}{x^{\frac{1}{6}}}\)[/tex] step-by-step.
### Step 1: Apply the Rules of Exponents
When dividing like bases, you subtract the exponents from each other. So, we start with:
[tex]\[
\frac{x^{\frac{5}{6}}}{x^{\frac{1}{6}}} = x^{\frac{5}{6} - \frac{1}{6}}
\][/tex]
### Step 2: Subtract the Exponents
Next, we need to subtract the exponents [tex]\(\frac{5}{6}\)[/tex] and [tex]\(\frac{1}{6}\)[/tex]:
[tex]\[
\frac{5}{6} - \frac{1}{6} = \frac{5 - 1}{6} = \frac{4}{6} = \frac{2}{3}
\][/tex]
### Step 3: Rewrite the Expression with the Simplified Exponent
Now that we have simplified the exponent, the expression becomes:
[tex]\[
x^{\frac{2}{3}}
\][/tex]
### Step 4: Convert to Radical Form
To convert [tex]\(x^{\frac{2}{3}}\)[/tex] into radical form, recall that [tex]\(a^{m/n}\)[/tex] can be written as [tex]\(\sqrt[n]{a^m}\)[/tex]. Thus, [tex]\(x^{\frac{2}{3}}\)[/tex] can be written as:
[tex]\[
x^{\frac{2}{3}} = \sqrt[3]{x^2}
\][/tex]
### Final Answer
The expression [tex]\(\frac{x^{\frac{5}{6}}}{x^{\frac{1}{6}}}\)[/tex] rewritten in simplest radical form is:
[tex]\[
\sqrt[3]{x^2}
\][/tex]
So, [tex]\(\frac{x^{\frac{5}{6}}}{x^{\frac{1}{6}}} = \sqrt[3]{x^2}\)[/tex].
