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

Video Solution

Solution Steps

00:00 Simplify the following problem

00:03 Every number is to the power of 1

00:12 When we have a root of order (B) on number (X) to the power of (A)

00:17 The result equals number (X) to the power of (A divided by B)

00:20 Apply this formula to our exercise

00:30 When we have a division of powers (A\B) with equal bases

00:35 The result equals the common base to the power of the difference of exponents (A - B)

00:38 Apply this formula to our exercise, and subtract between the powers

00:43 Determine the common denominator and calculate the power

00:58 This is the solution

Step-by-Step Solution

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 $36^{\frac{1}{10}}$ .