Solve Nested Square Roots: Simplifying √(√625) Step by Step

Q: Why can't I just take the fourth root of 625?

Great observation! While $ \sqrt{\sqrt{625}} = \sqrt[4]{625} $ mathematically, it's much easier to work step by step. Finding $ \sqrt{625} = 25 $ first is simpler than calculating $ \sqrt[4]{625} $ directly.

Q: How do I recognize that 625 is a perfect square?

Look for patterns! Since 625 ends in 25, try squares ending in 5: $ 15^2 = 225 $ (too small), $ 25^2 = 625 $ (perfect!). Practice with common perfect squares to build recognition.

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

You'd leave it in radical form! For example, $ \sqrt{\sqrt{50}} = \sqrt{5\sqrt{2}} $. But problems like this usually use perfect squares to keep answers clean and simple.

Q: Can I use a calculator for this?

You can, but recognizing perfect squares mentally is faster! Knowing that $ 25^2 = 625 $ and $ 5^2 = 25 $ lets you solve $ \sqrt{\sqrt{625}} $ in seconds without technology.

Q: Why do we work from inside to outside?

Mathematical operations have a natural order! Just like parentheses in arithmetic, nested radicals must be evaluated from the innermost first. This ensures you get the correct result every time.

Nested Radicals with Perfect Square Recognition

Solve the following exercise:

$\sqrt{\sqrt{625}}=$

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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

00:10 When we have a number (A) in a root of the order (B) in a root of the order (C)

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

00:22 We'll apply this formula to our exercise

00:28 Calculate the order multiplication

00:38 When we have a number (A) raised to the power (B) in a root of order (C)

00:43 The result equals the number (A) raised to the power (B divided by C)

00:47 We'll apply this formula to our exercise

00:54 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:

$\sqrt{\sqrt{625}}=$

Step-by-step solution

To solve this problem, we'll follow these steps:

Step 1: Evaluate the innermost square root, $\sqrt{625}$ .
Step 2: Evaluate the square root of the result from Step 1.

Now, let's work through each step:

Step 1: Evaluate $\sqrt{625}$ .
The square root of 625 is 25, since $25 \times 25 = 625$ . Thus, $\sqrt{625} = 25$ .

Step 2: Evaluate $\sqrt{25}$ .
The square root of 25 is 5, since $5 \times 5 = 25$ . Thus, $\sqrt{25} = 5$ .

Therefore, the solution to the problem $\sqrt{\sqrt{625}}$ is 5.

Final Answer

Key Points to Remember

Essential concepts to master this topic

Order: Always evaluate the innermost radical first
Recognition: Identify perfect squares like $\sqrt{625} = 25$
Verify: Check that $5^2 = 25$ and $25^2 = 625$ ✓

Common Mistakes

Avoid these frequent errors

Trying to combine nested radicals without evaluating first
Don't try to simplify $\sqrt{\sqrt{625}}$ using properties like $\sqrt{a\sqrt{b}}$ = wrong approach! This leads to confusion and errors. Always work from the inside out, evaluating $\sqrt{625} = 25$ first, then $\sqrt{25} = 5$ .

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 take the fourth root of 625?

Great observation! While $\sqrt{\sqrt{625}} = \sqrt[4]{625}$ mathematically, it's much easier to work step by step. Finding $\sqrt{625} = 25$ first is simpler than calculating $\sqrt[4]{625}$ directly.

How do I recognize that 625 is a perfect square?

Look for patterns! Since 625 ends in 25, try squares ending in 5: $15^2 = 225$ (too small), $25^2 = 625$ (perfect!). Practice with common perfect squares to build recognition.

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

You'd leave it in radical form! For example, $\sqrt{\sqrt{50}} = \sqrt{5\sqrt{2}}$ . But problems like this usually use perfect squares to keep answers clean and simple.

Can I use a calculator for this?

You can, but recognizing perfect squares mentally is faster! Knowing that $25^2 = 625$ and $5^2 = 25$ lets you solve $\sqrt{\sqrt{625}}$ in seconds without technology.

Why do we work from inside to outside?

Mathematical operations have a natural order! Just like parentheses in arithmetic, nested radicals must be evaluated from the innermost first. This ensures you get the correct result every time.

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