Sure, let's simplify the given expression [tex]\(\frac{x^{\frac{5}{6}}}{x^{\frac{1}{6}}}\)[/tex] step-by-step.
### Step 1: Apply the properties of exponents
To simplify the expression [tex]\(\frac{x^{\frac{5}{6}}}{x^{\frac{1}{6}}}\)[/tex], we use the property of exponents that states:
[tex]\[
\frac{a^m}{a^n} = a^{m-n}
\][/tex]
In this case, our base [tex]\(a\)[/tex] is [tex]\(x\)[/tex], [tex]\(m\)[/tex] is [tex]\(\frac{5}{6}\)[/tex], and [tex]\(n\)[/tex] is [tex]\(\frac{1}{6}\)[/tex].
### Step 2: Subtract the exponents
We need to subtract [tex]\(\frac{1}{6}\)[/tex] from [tex]\(\frac{5}{6}\)[/tex]:
[tex]\[
\frac{5}{6} - \frac{1}{6} = \frac{5 - 1}{6} = \frac{4}{6}
\][/tex]
### Step 3: Simplify the fraction
Next, we simplify the fraction [tex]\(\frac{4}{6}\)[/tex]:
[tex]\[
\frac{4}{6} = \frac{2}{3}
\][/tex]
### Step 4: Rewrite the expression with the simplified exponent
After simplifying, the exponent becomes [tex]\(\frac{2}{3}\)[/tex]. Therefore, we can rewrite the expression as:
[tex]\[
x^{\frac{2}{3}}
\][/tex]
### Step 5: Convert to simplest radical form
To express [tex]\(x^{\frac{2}{3}}\)[/tex] in its simplest radical form, we recognize that the fraction [tex]\(\frac{2}{3}\)[/tex] can be interpreted as:
[tex]\[
x^{\frac{2}{3}} = (x^{\frac{1}{3}})^2
\][/tex]
The exponent [tex]\(\frac{1}{3}\)[/tex] represents the cube root, so we can rewrite this as:
[tex]\[
(x^{\frac{1}{3}})^2 = (\sqrt[3]{x})^2 = \sqrt[3]{x^2}
\][/tex]
Thus, the expression [tex]\(\frac{x^{\frac{5}{6}}}{x^{\frac{1}{6}}}\)[/tex] simplifies to:
[tex]\[
\sqrt[3]{x^2}
\][/tex]
So, the simplest radical form of [tex]\(\frac{x^{\frac{5}{6}}}{x^{\frac{1}{6}}}\)[/tex] is [tex]\(\sqrt[3]{x^2}\)[/tex].
