Simplify the Ratio: Fifth Root of 36 Divided by Tenth Root of 36

To solve this problem, we'll transform the given roots into expressions with fractional exponents and then simplify using the rules of exponents.

• Step 1: Express roots as fractional exponents - $\sqrt[5]{36} = 36^{1/5}$ - $\sqrt[10]{36} = 36^{1/10}$

• Step 2: Apply the quotient rule for exponents - We simplify $\frac{36^{1/5}}{36^{1/10}}$ using the property: $\frac{a^m}{a^n} = a^{m-n}$: $\frac{36^{1/5}}{36^{1/10}} = 36^{1/5 - 1/10}$

• Step 3: Simplify the exponent - First, find a common denominator for the exponents: $1/5 = 2/10$ - The subtraction gives us: $2/10 - 1/10 = 1/10$

Thus, the simplified expression is $36^{1/10}$.

Therefore, the solution to the problem is $36^{\frac{1}{10}}$.