Solve the Nested Root Expression: Fifth Root of Cube Root of 5

Q: Why do I convert roots to fractional exponents?

Converting to fractional exponents makes it much easier to apply exponent rules! $ \sqrt[n]{a} = a^{1/n} $ transforms the problem into simple multiplication of fractions.

Q: How do I multiply fractions like 1/3 × 1/5?

Multiply the numerators together and the denominators together: $ \frac{1}{3} \times \frac{1}{5} = \frac{1 \times 1}{3 \times 5} = \frac{1}{15} $

Q: Can I work from the outside root inward instead?

Yes, but it's more confusing! Starting with the innermost root $ \sqrt[3]{5} $ and working outward follows the natural order of operations and is less error-prone.

Q: How can I check if my final answer is correct?

Work backwards! If your answer is $ \sqrt[15]{5} $, then $ (\sqrt[15]{5})^{15} $ should equal 5. You can also use a calculator to verify both expressions give the same decimal value.

Nested Radical Expressions with Exponent Rules

Solve the following exercise:

$\sqrt[5]{\sqrt[3]{5}}=$

❤️ Continue Your Math Journey!

We have hundreds of course questions with personalized recommendations + Account 100% premium

Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Solve the following problem

00:04 When we have a number (A) in a root of the order (B) in a root of the order (C)

00:09 The result equals the number (A) in a root of the order of their product (B times C)

00:13 Let's apply this formula to our exercise

00:19 Calculate the order of multiplication

00:24 This is the solution

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

Solve the following exercise:

$\sqrt[5]{\sqrt[3]{5}}=$

Step-by-step solution

To solve the problem of finding $\sqrt[5]{\sqrt[3]{5}}$ , we'll use the formula for a root of a root, which combines the exponents:

Step 1: Express each root as an exponent.
We start with the innermost root: $\sqrt[3]{5} = 5^{1/3}$ .
Step 2: Apply the outer root.
The square root to the fifth power is expressed as: $\sqrt[5]{5^{1/3}} = (5^{1/3})^{1/5}$ .
Step 3: Combine the exponents.
Using the exponent rule $(a^m)^n = a^{m \times n}$ , we get:
$(5^{1/3})^{1/5} = 5^{(1/3) \times (1/5)} = 5^{1/15}$ .
Step 4: Convert the exponent back to root form.
This can be written as $\sqrt[15]{5}$ .

Therefore, the simplified expression of $\sqrt[5]{\sqrt[3]{5}}$ is $\sqrt[15]{5}$ .

Final Answer

$\sqrt[15]{5}$

Key Points to Remember

Essential concepts to master this topic

Exponent Rule: Convert nested roots to fractional exponents first
Technique: Multiply exponents: $(5^{1/3})^{1/5} = 5^{1/3 \times 1/5} = 5^{1/15}$
Check: Verify by working backwards: $(\sqrt[15]{5})^{15} = 5$ ✓

Common Mistakes

Avoid these frequent errors

Adding the root indices instead of multiplying exponents
Don't add 5 + 3 = 8 to get $\sqrt[8]{5}$ ! This ignores the nested structure and gives a completely wrong answer. Always convert to fractional exponents first, then multiply: 1/3 × 1/5 = 1/15.

Practice Quiz

Test your knowledge with interactive questions

Solve the following exercise:

$\sqrt[10]{\sqrt[10]{1}}=$

FAQ

Everything you need to know about this question

Why do I convert roots to fractional exponents?

Converting to fractional exponents makes it much easier to apply exponent rules! $\sqrt[n]{a} = a^{1/n}$ transforms the problem into simple multiplication of fractions.

How do I multiply fractions like 1/3 × 1/5?

Multiply the numerators together and the denominators together: $\frac{1}{3} \times \frac{1}{5} = \frac{1 \times 1}{3 \times 5} = \frac{1}{15}$

Can I work from the outside root inward instead?

Yes, but it's more confusing! Starting with the innermost root $\sqrt[3]{5}$ and working outward follows the natural order of operations and is less error-prone.

What if there were three nested roots?

Same process! Convert each root to a fractional exponent and multiply all the fractions. For example: $\sqrt[2]{\sqrt[3]{\sqrt[4]{5}}} = 5^{1/4 \times 1/3 \times 1/2} = 5^{1/24}$

How can I check if my final answer is correct?

Work backwards! If your answer is $\sqrt[15]{5}$ , then $(\sqrt[15]{5})^{15}$ should equal 5. You can also use a calculator to verify both expressions give the same decimal value.

🌟 Unlock Your Math Potential

Get unlimited access to all 18 Rules of Roots questions, detailed video solutions, and personalized progress tracking.

📹

Unlimited Video Solutions

Step-by-step explanations for every problem

📊

Progress Analytics

Track your mastery across all topics

🚫

Ad-Free Learning

Focus on math without distractions

No credit card required • Cancel anytime