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

Q: Why do I need to convert radicals to fractional exponents?

Converting makes division much easier! Fractional exponents let you use the simple quotient rule $ a^m ÷ a^n = a^{m-n} $ instead of complex radical division rules.

Q: How do I subtract fractions with different denominators?

Find a common denominator first! For $ \frac{1}{5} - \frac{1}{10} $, convert $ \frac{1}{5} $ to $ \frac{2}{10} $, then subtract: $ \frac{2}{10} - \frac{1}{10} = \frac{1}{10} $.

Q: Can I just cancel the 36s in the fraction?

No! You can only cancel when you have multiplication, not when dealing with different roots. The bases are the same, but the exponents are different, so use the quotient rule instead.

Q: What's the difference between the fifth root and tenth root?

The fifth root asks "what number multiplied by itself 5 times equals 36?" The tenth root asks the same but with 10 times. Since 10 > 5, the tenth root gives a smaller result.

Radical Division with Fractional Exponents

Solve the following exercise:

$\frac{\sqrt[5]{36}}{\sqrt[10]{36}}=$

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

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 written solution

Follow each step carefully to understand the complete solution

Understand the problem

Solve the following exercise:

$\frac{\sqrt[5]{36}}{\sqrt[10]{36}}=$

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

Final Answer

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

Key Points to Remember

Essential concepts to master this topic

Rule: Convert radicals to fractional exponents before dividing
Technique: Use quotient rule $a^m ÷ a^n = a^{m-n}$ with common denominators
Check: Verify $36^{1/10}$ equals tenth root of 36 ✓

Common Mistakes

Avoid these frequent errors

Subtracting denominators instead of fractions
Don't subtract 10 - 5 = 5 to get $36^{1/5}$ ! This ignores fraction subtraction rules and gives the wrong exponent. Always convert to common denominators: $\frac{1}{5} - \frac{1}{10} = \frac{2}{10} - \frac{1}{10} = \frac{1}{10}$ .

Practice Quiz

Test your knowledge with interactive questions

Solve the following exercise:

$\sqrt{\frac{2}{4}}=$

FAQ

Everything you need to know about this question

Why do I need to convert radicals to fractional exponents?

Converting makes division much easier! Fractional exponents let you use the simple quotient rule $a^m ÷ a^n = a^{m-n}$ instead of complex radical division rules.

How do I subtract fractions with different denominators?

Find a common denominator first! For $\frac{1}{5} - \frac{1}{10}$ , convert $\frac{1}{5}$ to $\frac{2}{10}$ , then subtract: $\frac{2}{10} - \frac{1}{10} = \frac{1}{10}$ .

Can I just cancel the 36s in the fraction?

No! You can only cancel when you have multiplication, not when dealing with different roots. The bases are the same, but the exponents are different, so use the quotient rule instead.

What's the difference between the fifth root and tenth root?

The fifth root asks "what number multiplied by itself 5 times equals 36?" The tenth root asks the same but with 10 times. Since 10 > 5, the tenth root gives a smaller result.

How can I check if my answer is correct?

Multiply your answer by itself according to the original problem! Since we got $36^{1/10}$ , verify that $36^{1/5} ÷ 36^{1/10} = 36^{1/10}$ using a calculator.

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