Solve the Nested Radical: Finding √√2

Q: Why can't I just write this as √2?

Because √√2 is different from √2! Think of it this way: √√2 means "the square root of the square root of 2", which requires two square root operations, not just one.

Q: How do I remember when to multiply vs add exponents?

Multiply exponents when taking a power of a power: $ (x^a)^b = x^{ab} $. Add exponents when multiplying same bases: $ x^a \cdot x^b = x^{a+b} $.

Q: What does the fourth root symbol mean?

The fourth root $ \sqrt[4]{2} $ asks: "What number, when multiplied by itself 4 times, equals 2?" It's the opposite operation of raising to the 4th power.

Q: Can I check my answer without a calculator?

Yes! If $ \sqrt{\sqrt{2}} = \sqrt[4]{2} $ is correct, then squaring both sides should give $ \sqrt{2} = \sqrt{2} $. Try it: $ (\sqrt[4]{2})^2 = 2^{1/2} = \sqrt{2} $ ✓

Q: Are there other ways to write this answer?

Absolutely! $ \sqrt[4]{2} $, $ 2^{1/4} $, and $ 2^{0.25} $ are all equivalent. The fourth root notation is usually preferred for clarity.

Radical Properties with Nested Square Roots

Solve the following exercise:

$\sqrt{\sqrt{2}}=$

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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) under a root of the order (B) under a root of the order (C)

00:14 The result equals the number (A) under a root of the order of their product (B times C)

00:18 Apply this formula to our exercise

00:23 Calculate the order of multiplication

00:29 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:

$\sqrt{\sqrt{2}}=$

Step-by-step solution

To solve $\sqrt{\sqrt{2}}$ , we will use the property of roots.

Step 1: Recognize that $\sqrt{\sqrt{2}}$ involves two square roots.
Step 2: Each square root can be expressed using exponents: $\sqrt{2} = 2^{1/2}$ .
Step 3: Therefore, $\sqrt{\sqrt{2}} = (2^{1/2})^{1/2}$ .
Step 4: Apply the formula for the root of a root: $(x^{a})^{b} = x^{ab}$ .
Step 5: For $(2^{1/2})^{1/2}$ , this means we compute the product of the exponents: $(1/2) \times (1/2) = 1/4$ .
Step 6: The expression simplifies to $2^{1/4}$ , which is written as $\sqrt[4]{2}$ .

Therefore, $\sqrt{\sqrt{2}} = \sqrt[4]{2}$ .

This corresponds to choice 2: $\sqrt[4]{2}$ .

The solution to the problem is $\sqrt[4]{2}$ .

Final Answer

$\sqrt[4]{2}$

Key Points to Remember

Essential concepts to master this topic

Rule: Nested radicals combine using the exponent multiplication property
Technique: Convert $\sqrt{\sqrt{2}}$ to $(2^{1/2})^{1/2} = 2^{1/4}$
Check: Verify $(\sqrt[4]{2})^4 = 2$ and $(\sqrt[4]{2})^2 = \sqrt{2}$ ✓

Common Mistakes

Avoid these frequent errors

Adding exponents instead of multiplying them
Don't add the exponents: 1/2 + 1/2 = 1 gives $\sqrt{2}$ = wrong answer! This ignores the nested structure of the radicals. Always multiply exponents when taking a root of a root: (1/2) × (1/2) = 1/4.

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 √2?

Because √√2 is different from √2! Think of it this way: √√2 means "the square root of the square root of 2", which requires two square root operations, not just one.

How do I remember when to multiply vs add exponents?

Multiply exponents when taking a power of a power: $(x^a)^b = x^{ab}$ . Add exponents when multiplying same bases: $x^a \cdot x^b = x^{a+b}$ .

What does the fourth root symbol mean?

The fourth root $\sqrt[4]{2}$ asks: "What number, when multiplied by itself 4 times, equals 2?" It's the opposite operation of raising to the 4th power.

Can I check my answer without a calculator?

Yes! If $\sqrt{\sqrt{2}} = \sqrt[4]{2}$ is correct, then squaring both sides should give $\sqrt{2} = \sqrt{2}$ . Try it: $(\sqrt[4]{2})^2 = 2^{1/2} = \sqrt{2}$ ✓

Are there other ways to write this answer?

Absolutely! $\sqrt[4]{2}$ , $2^{1/4}$ , and $2^{0.25}$ are all equivalent. The fourth root notation is usually preferred for clarity.

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