Simplify the Nested Radical: Fifth Root of Square Root of x^20

Q: Why do I convert radicals to fractional exponents?

Fractional exponents make nested radicals much easier to work with! $ \sqrt[n]{x} = x^{\frac{1}{n}} $ lets you use the power rule: $ (x^a)^b = x^{ab} $.

Q: How do I handle the nested structure?

Work from the inside out! First simplify $ \sqrt{x^{20}} = x^{10} $, then take the fifth root: $ \sqrt[5]{x^{10}} = x^2 $.

Q: Can I combine the fractional exponents directly?

Yes! You can multiply the fractional exponents: $ x^{20 \cdot \frac{1}{2} \cdot \frac{1}{5}} = x^{20 \cdot \frac{1}{10}} = x^2 $. This gives the same answer faster!

Q: What if the inner exponent doesn't divide evenly?

No problem! Even if you get something like $ x^{\frac{7}{3}} $, just leave it as a fractional exponent. Not all expressions simplify to whole number powers.

Q: How do I check if my answer is right?

Work backwards! If your answer is $ x^2 $, then $ (x^2)^5 = x^{10} $ and $ (x^{10})^2 = x^{20} $. This confirms your simplification is correct!

Fractional Exponents with Nested Radicals

Complete the following exercise:

$\sqrt[5]{\sqrt{x^{20}}}=$

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Solve the following problem

00:03 A "regular" root is of the order 2

00:09 When we have a number (A) raised to the power (B) with the root order (C)

00:19 The result equals the number (A) with the root order of their product (B times C)

00:24 We will apply this formula to our exercise

00:31 Let's calculate the multiplication of the orders

00:41 When we have a number (A) raised to the power (B) with the root order (C)

00:46 The result equals the number (A) raised to the power of their quotient (B divided by C)

00:49 We will apply this formula to our exercise

00:53 Proceed to calculate the division of powers

00:57 This is the solution

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

Complete the following exercise:

$\sqrt[5]{\sqrt{x^{20}}}=$

Step-by-step solution

To solve the problem $\sqrt[5]{\sqrt{x^{20}}}$ , we will convert the roots into fractional exponents and simplify:

The expression $\sqrt{x^{20}}$ can be represented as:

$(x^{20})^{\frac{1}{2}} = x^{20 \cdot \frac{1}{2}} = x^{10}$

Now, taking the fifth root of $x^{10}$ , we have:

$(x^{10})^{\frac{1}{5}} = x^{10 \cdot \frac{1}{5}} = x^{2}$

Therefore, the original expression $\sqrt[5]{\sqrt{x^{20}}}$ simplifies to $x^2$ .

Final Answer

$x^2$

Key Points to Remember

Essential concepts to master this topic

Rule: Convert nested radicals to fractional exponents first
Technique: $\sqrt{x^{20}} = x^{10}$ , then $\sqrt[5]{x^{10}} = x^{2}$
Check: Verify that $(x^2)^5 = x^{10}$ and $(x^{10})^2 = x^{20}$ ✓

Common Mistakes

Avoid these frequent errors

Adding exponents instead of multiplying
Don't think $\sqrt[5]{\sqrt{x^{20}}} = x^{20+\frac{1}{2}+\frac{1}{5}}$ ! This completely ignores how exponent rules work. Always multiply exponents: $x^{20} \cdot \frac{1}{2} \cdot \frac{1}{5} = x^2$ .

Practice Quiz

Test your knowledge with interactive questions

Solve the following exercise:

$\sqrt{\sqrt{4}}=$

FAQ

Everything you need to know about this question

Why do I convert radicals to fractional exponents?

Fractional exponents make nested radicals much easier to work with! $\sqrt[n]{x} = x^{\frac{1}{n}}$ lets you use the power rule: $(x^a)^b = x^{ab}$ .

How do I handle the nested structure?

Work from the inside out! First simplify $\sqrt{x^{20}} = x^{10}$ , then take the fifth root: $\sqrt[5]{x^{10}} = x^2$ .

Can I combine the fractional exponents directly?

Yes! You can multiply the fractional exponents: $x^{20 \cdot \frac{1}{2} \cdot \frac{1}{5}} = x^{20 \cdot \frac{1}{10}} = x^2$ . This gives the same answer faster!

What if the inner exponent doesn't divide evenly?

No problem! Even if you get something like $x^{\frac{7}{3}}$ , just leave it as a fractional exponent. Not all expressions simplify to whole number powers.

How do I check if my answer is right?

Work backwards! If your answer is $x^2$ , then $(x^2)^5 = x^{10}$ and $(x^{10})^2 = x^{20}$ . This confirms your simplification is correct!

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