Question

simplify.
\\(\frac{\sqrt3{x^2}}{\sqrt6{x^3}}\\)
\\(x^{\frac{?}{?}}\\)

Explanation:

Step1: Convert radicals to exponents

Recall that \(\sqrt[n]{a^m}=a^{\frac{m}{n}}\). So, \(\sqrt[3]{x^2}=x^{\frac{2}{3}}\) and \(\sqrt[6]{x^3}=x^{\frac{3}{6}} = x^{\frac{1}{2}}\).
The expression becomes \(\frac{x^{\frac{2}{3}}}{x^{\frac{1}{2}}}\).

Step2: Use exponent rule for division

When dividing with the same base, subtract the exponents: \(a^m\div a^n=a^{m - n}\).
So, \(x^{\frac{2}{3}-\frac{1}{2}}\).

Step3: Subtract the fractions

Find a common denominator for \(\frac{2}{3}\) and \(\frac{1}{2}\), which is 6.
\(\frac{2}{3}=\frac{4}{6}\) and \(\frac{1}{2}=\frac{3}{6}\).
Then \(\frac{4}{6}-\frac{3}{6}=\frac{1}{6}\).
So the exponent is \(\frac{1}{6}\), and the expression simplifies to \(x^{\frac{1}{6}}\), which is also \(\sqrt[6]{x}\), but in the form \(x^{\frac{1}{6}}\), the exponent is \(\frac{1}{6}\).

Answer:

\(\frac{1}{6}\) (so the expression is \(x^{\frac{1}{6}}\))