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

Video Solution

Solution Steps

00:08 Let's simplify this problem.

00:11 Remember, every number has a power of one.

00:20 If you see a root of order B on X to the power of A,

00:25 it equals X to the power of A divided by B.

00:30 Now, let's use this in our exercise.

00:38 When dividing powers with the same base,

00:43 we get the base to the power of A minus B.

00:47 Apply this by subtracting the exponents.

00:51 Find the common denominator, then calculate the power.

01:06 And that's how we solve this problem!

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