Simplify the Nested Radical: 10th Root of Square Root of x^20

Q: Why do I convert radicals to exponents?

Exponent notation makes it much easier to apply power rules! $ \sqrt{x^{20}} $ becomes $ (x^{20})^{1/2} $, and $ \sqrt[10]{} $ becomes $ ()^{1/10} $.

Q: What's the difference between adding and multiplying exponents?

You add exponents when multiplying same bases: $ x^2 \cdot x^3 = x^5 $. You multiply exponents for power of a power: $ (x^2)^3 = x^6 $. Nested radicals use the second rule!

Q: Can I simplify this a different way?

Yes! You could also think: $ \sqrt{x^{20}} = x^{10} $ (since we take half the exponent), then $ \sqrt[10]{x^{10}} = x $ (since the 10th root and 10th power cancel).

Q: What if x is negative?

For this problem, we assume $ x ≥ 0 $ so all roots are defined. With negative values, even roots like square roots become more complex and may not be real numbers.

Q: How do I check my answer is right?

Substitute back! If the answer is $ x $, then $ \sqrt[10]{\sqrt{x^{20}}} $ should equal $ x $. Work backwards: $ x^{10} $ under the square root gives $ x^{20} $ ✓

Radical Simplification with Nested Roots

Comlete the following exercise:

$\sqrt[10]{\sqrt{x^{20}}}=$

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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

00:11 When we have a number (A) raised to the power (B) in a root of order (C)

00:17 The result equals the number (A) in a root of order of their product (B times C)

00:21 We will apply this formula to our exercise

00:25 Calculate the multiplication of orders

00:34 When we have a number (A) raised to the power (B) in a root of order (C)

00:40 The result equals the number (A) raised to the power of their quotient (B divided by C)

00:43 We will apply this formula to our exercise

00:49 Calculate the quotient of the powers

00:52 This is the solution

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

Comlete the following exercise:

$\sqrt[10]{\sqrt{x^{20}}}=$

Step-by-step solution

To solve the problem, we'll simplify $\sqrt[10]{\sqrt{x^{20}}}$ using properties of exponents and roots:

Step 1: Convert each part into exponent form. We have $\sqrt{x^{20}}$ , which can be rewritten as $(x^{20})^{1/2}$ .
Step 2: Simplify the expression $(x^{20})^{1/2}$ . Using the power of a power property, this becomes $x^{20 \cdot (1/2)} = x^{10}$ .
Step 3: Apply the 10th root to the simplified result: $\sqrt[10]{x^{10}}$ , which can be written as $(x^{10})^{1/10}$ .
Step 4: Simplify $(x^{10})^{1/10}$ using the power of a power property again, we get $x^{(10 \cdot (1/10))} = x^1 = x$ .

Therefore, the expression simplifies to $x$ .

Conclusion: The solution to the problem is $x$ .

Final Answer

$x$

Key Points to Remember

Essential concepts to master this topic

Exponent Rule: Convert radicals to fractional exponents for easier manipulation
Power Property: Use $(a^m)^n = a^{mn}$ to simplify nested expressions
Verification: Check that $x^1 = x$ makes sense in original form ✓

Common Mistakes

Avoid these frequent errors

Adding exponents instead of multiplying
Don't add the exponents 20 + 1/2 + 1/10 = 20.6! This ignores the proper order of operations for nested radicals. Always work from inside out: first simplify $x^{20 \cdot 1/2} = x^{10}$ , then apply $x^{10 \cdot 1/10} = x^1$ .

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 radicals to exponents?

Exponent notation makes it much easier to apply power rules! $\sqrt{x^{20}}$ becomes $(x^{20})^{1/2}$ , and $\sqrt[10]{}$ becomes $()^{1/10}$ .

What's the difference between adding and multiplying exponents?

You add exponents when multiplying same bases: $x^2 \cdot x^3 = x^5$ . You multiply exponents for power of a power: $(x^2)^3 = x^6$ . Nested radicals use the second rule!

Can I simplify this a different way?

Yes! You could also think: $\sqrt{x^{20}} = x^{10}$ (since we take half the exponent), then $\sqrt[10]{x^{10}} = x$ (since the 10th root and 10th power cancel).

What if x is negative?

For this problem, we assume $x ≥ 0$ so all roots are defined. With negative values, even roots like square roots become more complex and may not be real numbers.

How do I check my answer is right?

Substitute back! If the answer is $x$ , then $\sqrt[10]{\sqrt{x^{20}}}$ should equal $x$ . Work backwards: $x^{10}$ under the square root gives $x^{20}$ ✓

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