To simplify the expression [tex]\(\frac{x^{\frac{5}{6}}}{x^{\frac{1}{6}}}\)[/tex] in its simplest radical form, follow these steps:
1. Identify the Expression: The given expression is [tex]\(\frac{x^{\frac{5}{6}}}{x^{\frac{1}{6}}}\)[/tex].
2. Apply the Quotient Rule for Exponents: The Quotient Rule states that [tex]\(\frac{a^m}{a^n} = a^{m-n}\)[/tex], where [tex]\(a\)[/tex] is a nonzero real number, and [tex]\(m\)[/tex] and [tex]\(n\)[/tex] are any real numbers.
[tex]\[
\frac{x^{\frac{5}{6}}}{x^{\frac{1}{6}}} = x^{\frac{5}{6} - \frac{1}{6}}
\][/tex]
3. Subtract the Exponents: 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]
4. Simplify the Fraction: Simplify the fraction [tex]\(\frac{4}{6}\)[/tex] by dividing both the numerator and the denominator by their greatest common divisor, which is 2.
[tex]\[
\frac{4}{6} = \frac{4 \div 2}{6 \div 2} = \frac{2}{3}
\][/tex]
5. Write the Simplified Exponent: Now, the expression is simplified to:
[tex]\[
x^{\frac{2}{3}}
\][/tex]
6. Convert to Radical Form: To rewrite [tex]\(x^{\frac{2}{3}}\)[/tex] in radical form, recall that [tex]\(a^{\frac{m}{n}} = \sqrt[n]{a^m}\)[/tex]. Here, [tex]\(a = x\)[/tex], [tex]\(m = 2\)[/tex], and [tex]\(n = 3\)[/tex].
[tex]\[
x^{\frac{2}{3}} = \sqrt[3]{x^2}
\][/tex]
Therefore, the simplest radical form of the given expression [tex]\(\frac{x^{\frac{5}{6}}}{x^{\frac{1}{6}}}\)[/tex] is [tex]\(\sqrt[3]{x^2}\)[/tex].
