Solve the Nested Square Root: Finding √√4

Q: Why can't I just write this as a fourth root?

While $ \sqrt{\sqrt{4}} $ does equal $ 4^{1/4} $, it's easier to work step by step. Evaluate the inner square root first: $ \sqrt{4} = 2 $, then find $ \sqrt{2} $.

Q: How do I know when to stop simplifying?

Stop when you can't simplify further! Since 2 is a prime number, $ \sqrt{2} $ cannot be simplified to a whole number. This is your final answer.

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

If the inner radical doesn't simplify to a whole number, leave it as is and then take the square root of that expression. For example: $ \sqrt{\sqrt{3}} $ stays as $ \sqrt{\sqrt{3}} $.

Q: Can I use a calculator for nested radicals?

Yes, but work step by step to understand the process! Calculate $ \sqrt{4} = 2 $ first, then $ \sqrt{2} ≈ 1.414 $. This helps you see the logic behind nested radicals.

Q: Are there other ways to write the final answer?

The exact answer is $ \sqrt{2} $. You could also write it as $ 2^{1/2} $ or $ 4^{1/4} $, but $ \sqrt{2} $ is the simplest radical form.

Nested Radicals with Inner Evaluation

Solve the following exercise:

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

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:06 Let's solve this math problem together.

00:09 A regular square root is also known as a root of order two.

00:16 We'll express four as two to the power of two.

00:21 When a number, let's call it A, is raised to the power B under a root of order B.

00:28 The root and the power cancel each other out, leaving us with the number A.

00:32 Let's use this rule in our exercise now.

00:36 And that's how we find the solution!

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

Solve the following exercise:

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

Step-by-step solution

To solve the expression $\sqrt{\sqrt{4}}$ , we'll proceed with the following steps:

Step 1: Evaluate the inner square root.
The expression $\sqrt{4}$ simplifies to 2, because 2 squared is 4.
Step 2: Now evaluate the square root of 2.
Since the result from step 1 is 2, we need to find $\sqrt{2}$ . This is the prime representation of the result because 2 cannot be further simplified.

Therefore, the answer to the problem $\sqrt{\sqrt{4}}$ is $\sqrt{2}$ .

Final Answer

$\sqrt{2}$

Key Points to Remember

Essential concepts to master this topic

Order Rule: Always evaluate the innermost radical first before outer operations
Technique: $\sqrt{4} = 2$ , then $\sqrt{2}$ for final answer
Check: Verify $(\sqrt{2})^2 = 2$ and $2^2 = 4$ ✓

Common Mistakes

Avoid these frequent errors

Trying to simplify the entire expression at once
Don't attempt $\sqrt{\sqrt{4}} = \sqrt[4]{4}$ directly = wrong approach! This skips the crucial step-by-step evaluation and leads to confusion. Always work from the inside out, evaluating $\sqrt{4} = 2$ first.

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 can't I just write this as a fourth root?

While $\sqrt{\sqrt{4}}$ does equal $4^{1/4}$ , it's easier to work step by step. Evaluate the inner square root first: $\sqrt{4} = 2$ , then find $\sqrt{2}$ .

How do I know when to stop simplifying?

Stop when you can't simplify further! Since 2 is a prime number, $\sqrt{2}$ cannot be simplified to a whole number. This is your final answer.

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

If the inner radical doesn't simplify to a whole number, leave it as is and then take the square root of that expression. For example: $\sqrt{\sqrt{3}}$ stays as $\sqrt{\sqrt{3}}$ .

Can I use a calculator for nested radicals?

Yes, but work step by step to understand the process! Calculate $\sqrt{4} = 2$ first, then $\sqrt{2} ≈ 1.414$ . This helps you see the logic behind nested radicals.

Are there other ways to write the final answer?

The exact answer is $\sqrt{2}$ . You could also write it as $2^{1/2}$ or $4^{1/4}$ , but $\sqrt{2}$ is the simplest radical form.

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