Value Comparison Exercise: Finding the Largest Among Given Numbers

Q: How do I convert nested radicals to exponential form?

Use the rule that nested operations multiply exponents. For $ \sqrt{\sqrt{2}} $, the inner $ \sqrt{2} = 2^{1/2} $, then the outer square root gives $ (2^{1/2})^{1/2} = 2^{1/4} $.

Q: Why is a larger fractional exponent a larger value?

When the base is greater than 1 (like 2), larger exponents give larger results. Since $ \frac{1}{4} > \frac{1}{6} $, we have $ 2^{1/4} > 2^{1/6} $.

Q: Can I use decimal approximations to compare?

Yes! Calculate: $ 2^{1/4} ≈ 1.189 $, $ 2^{1/6} ≈ 1.122 $, $ 2^{1/8} ≈ 1.091 $, $ 2^{1/10} ≈ 1.072 $. This confirms our fractional comparison!

Q: What if the question had different nested roots like cube roots?

Same process! For $ \sqrt[3]{\sqrt[4]{2}} $, convert step by step: $ \sqrt[4]{2} = 2^{1/4} $, then $ (2^{1/4})^{1/3} = 2^{1/12} $.

Q: How do I remember which fractional exponent is larger?

Think of fractions with the same numerator: smaller denominators mean larger fractions. So $ \frac{1}{4} > \frac{1}{6} > \frac{1}{8} > \frac{1}{10} $.

Nested Radical Exponents with Power Comparison

Choose the largest value:

❤️ Continue Your Math Journey!

We have hundreds of course questions with personalized recommendations + Account 100% premium

Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Choose the largest value

00:03 A 'regular' root raised to the second power

00:07 Combine into one root by multiplying the orders

00:10 A root raised to the 4th power is like a power with the inverse of 4

00:13 Apply the same method to the following expressions in order to determine the largest value

00:50 This is the solution

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

Choose the largest value:

Step-by-step solution

To solve this problem, we need to express each nested root as a power of 2:

For $\sqrt{\sqrt{2}}$ :
$\sqrt{\sqrt{2}} = 2^{1/4}$
For $\sqrt[3]{\sqrt{2}}$ :
$\sqrt[3]{\sqrt{2}} = 2^{1/6}$
For $\sqrt[4]{\sqrt{2}}$ :
$\sqrt[4]{\sqrt{2}} = 2^{1/8}$
For $\sqrt[5]{\sqrt{2}}$ :
$\sqrt[5]{\sqrt{2}} = 2^{1/10}$

Now, we compare these powers:

$2^{1/4} > 2^{1/6} > 2^{1/8} > 2^{1/10}$ .

Therefore, the largest value is $\sqrt{\sqrt{2}}$ , which corresponds to choice 1.

Final Answer

$\sqrt{\sqrt{2}}$

Key Points to Remember

Essential concepts to master this topic

Rule: Convert nested radicals to exponential form with fractional exponents
Technique: $\sqrt{\sqrt{2}} = 2^{1/2 \cdot 1/2} = 2^{1/4}$ using exponent multiplication
Check: Compare fractional exponents: 1/4 > 1/6 > 1/8 > 1/10 ✓

Common Mistakes

Avoid these frequent errors

Confusing nested root conversion
Don't convert $\sqrt{\sqrt{2}}$ to $2^{1/2}$ = wrong exponent! This ignores the outer root and gives an incorrect comparison. Always multiply the exponents: $\sqrt{\sqrt{2}} = 2^{1/2 \cdot 1/2} = 2^{1/4}$ .

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

How do I convert nested radicals to exponential form?

Use the rule that nested operations multiply exponents. For $\sqrt{\sqrt{2}}$ , the inner $\sqrt{2} = 2^{1/2}$ , then the outer square root gives $(2^{1/2})^{1/2} = 2^{1/4}$ .

Why is a larger fractional exponent a larger value?

When the base is greater than 1 (like 2), larger exponents give larger results. Since $\frac{1}{4} > \frac{1}{6}$ , we have $2^{1/4} > 2^{1/6}$ .

Can I use decimal approximations to compare?

Yes! Calculate: $2^{1/4} ≈ 1.189$ , $2^{1/6} ≈ 1.122$ , $2^{1/8} ≈ 1.091$ , $2^{1/10} ≈ 1.072$ . This confirms our fractional comparison!

What if the question had different nested roots like cube roots?

Same process! For $\sqrt[3]{\sqrt[4]{2}}$ , convert step by step: $\sqrt[4]{2} = 2^{1/4}$ , then $(2^{1/4})^{1/3} = 2^{1/12}$ .

How do I remember which fractional exponent is larger?

Think of fractions with the same numerator: smaller denominators mean larger fractions. So $\frac{1}{4} > \frac{1}{6} > \frac{1}{8} > \frac{1}{10}$ .

🌟 Unlock Your Math Potential

Get unlimited access to all 18 Rules of Roots questions, detailed video solutions, and personalized progress tracking.

📹

Unlimited Video Solutions

Step-by-step explanations for every problem

📊

Progress Analytics

Track your mastery across all topics

🚫

Ad-Free Learning

Focus on math without distractions

No credit card required • Cancel anytime