Simplify the Nested Radical: Sixth Root of Square Root of x^12

Q: Why do I need to simplify the inner radical first?

Working inside out is the safest approach! By simplifying $ \sqrt{x^{12}} = x^6 $ first, you get a cleaner expression that's easier to handle in the next step.

Q: What if the inner exponent wasn't a perfect square?

If you had something like $ \sqrt{x^{10}} $, you'd get $ x^5 $ inside, then $ \sqrt[6]{x^5} = x^{5/6} $. The method stays the same even with non-perfect results!

Q: How do I know when nested radicals simplify to just x?

Look for patterns! When the product of all fractional exponents equals 1, you get $ x^1 = x $. Here: $ \frac{12}{2} \times \frac{1}{6} = 6 \times \frac{1}{6} = 1 $.

Q: Can I work from outside to inside instead?

While possible, it's much trickier and more error-prone. Stick with the inside-out approach - it's the standard method that works reliably for all nested radical problems.

Nested Radicals with Exponential Simplification

Complete the following exercise:

$\sqrt[6]{\sqrt{x^{12}}}=$

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:07 Let's solve this problem together.

00:10 When we talk about a regular root, we mean a square root.

00:16 If we have A raised to the power B, inside a root of order C.

00:22 The answer is A to the power of B times C inside the root.

00:27 Now, let's use this formula in our exercise.

00:31 First, multiply the order of the root with the power.

00:38 Again, if it's A to the power B in a root order C.

00:42 The result is A to the power of B divided by C.

00:47 Let's try this formula in our example.

00:50 Now, divide the power by the root's order.

00:54 And that's how we find the solution. Great job!

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

Complete the following exercise:

$\sqrt[6]{\sqrt{x^{12}}}=$

Step-by-step solution

To solve $\sqrt[6]{\sqrt{x^{12}}}$ , we will follow these steps:

Step 1: Simplify the inner radical expression $\sqrt{x^{12}}$ .
Step 2: Use the property of roots, expressing $\sqrt{x^{12}}$ as a power of $x$ .
Step 3: Use the result from step 1 in the outer root expression.
Step 4: Simplify the entire expression using exponent rules.

Now, let's perform each of these steps:

Step 1: Simplify $\sqrt{x^{12}}$ .
$\sqrt{x^{12}} = x^{12/2} = x^6$ .

Step 2: Simplify the outer expression $\sqrt[6]{x^6}$ .
$\sqrt[6]{x^6} = (x^6)^{1/6}$ .

Step 3: Apply the exponent rule $(a^m)^{n} = a^{m \times n}$ .
$(x^6)^{1/6} = x^{6 \times 1/6} = x^1 = x$ .

Therefore, the simplified expression is $\boxed{x}$ .

Thus, the solution to $\sqrt[6]{\sqrt{x^{12}}}$ is $x$ .

Final Answer

$x$

Key Points to Remember

Essential concepts to master this topic

Rule: Convert nested radicals to fractional exponents for easier manipulation
Technique: Work inside out: $\sqrt{x^{12}} = x^6$ , then $\sqrt[6]{x^6} = x$
Check: Verify by reversing: $x \to x^6 \to \sqrt{x^{12}} \to \sqrt[6]{\sqrt{x^{12}}}$ ✓

Common Mistakes

Avoid these frequent errors

Combining the radical indexes incorrectly
Don't multiply the indexes and write $\sqrt[6 \times 2]{x^{12}} = \sqrt[12]{x^{12}} = x$ directly! This skips the crucial inside-out simplification and can lead to confusion with more complex expressions. Always simplify the innermost radical first, then work outward step by step.

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 need to simplify the inner radical first?

Working inside out is the safest approach! By simplifying $\sqrt{x^{12}} = x^6$ first, you get a cleaner expression that's easier to handle in the next step.

Can I use fractional exponents instead of radicals?

Absolutely! Converting to exponents often makes the work clearer: $\sqrt[6]{\sqrt{x^{12}}} = (x^{12})^{1/2 \cdot 1/6} = x^{12/12} = x$ . Both methods give the same answer.

What if the inner exponent wasn't a perfect square?

If you had something like $\sqrt{x^{10}}$ , you'd get $x^5$ inside, then $\sqrt[6]{x^5} = x^{5/6}$ . The method stays the same even with non-perfect results!

How do I know when nested radicals simplify to just x?

Look for patterns! When the product of all fractional exponents equals 1, you get $x^1 = x$ . Here: $\frac{12}{2} \times \frac{1}{6} = 6 \times \frac{1}{6} = 1$ .

Can I work from outside to inside instead?

While possible, it's much trickier and more error-prone. Stick with the inside-out approach - it's the standard method that works reliably for all nested radical problems.

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