Finding the Greatest Value: Mathematical Comparison Exercise

Q: How do I convert nested radicals to fractional exponents?

Work from the inside out! For $ \sqrt[6]{\sqrt{5}} $, first write $ \sqrt{5} = 5^{1/2} $, then $ \sqrt[6]{5^{1/2}} = (5^{1/2})^{1/6} $.

Q: What's the power rule for exponents again?

When you have a power raised to another power, multiply the exponents: $ (a^m)^n = a^{m \cdot n} $. So $ (5^{1/2})^{1/6} = 5^{1/2 \cdot 1/6} = 5^{1/12} $.

Q: Why can't I just estimate which radical looks bigger?

Visual estimation fails with complex radicals! $ \sqrt[6]{\sqrt{5}} $ and $ \sqrt[12]{5} $ look different but are actually equal. Always convert to fractional exponents to compare accurately.

Q: How do I handle a radical inside another radical?

Use the chain rule approach: Convert the innermost radical first, then work outward. For $ \sqrt[4]{\sqrt[3]{5}} $, start with $ \sqrt[3]{5} = 5^{1/3} $, then $ \sqrt[4]{5^{1/3}} = 5^{1/12} $.

Q: What if the expressions aren't equal?

Then you'd compare the fractional exponents! Since $ a^m > a^n $ when $ a > 1 $ and $ m > n $, the expression with the larger exponent would be greatest.

Q: Can I use a calculator to check my work?

Absolutely! Calculate each expression's decimal value. If they're all equal (like 1.1487...), then all values are equal. This confirms your algebraic work!

Radical Expressions with Equivalent Powers

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 together

00:11 This is the value of the first expression

00:19 Apply the same method to this expression

00:24 They are all equal, and 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 determine if one of these values is the largest or if they are equal, we will express each expression as a power of 5:

First, consider $\sqrt[6]{\sqrt{5}}$ . This is equivalent to $(\sqrt{5})^{1/6} = (5^{1/2})^{1/6}$ . Applying the power rule, this is $5^{1/12}$ .
Next, consider $\sqrt[12]{5}$ , which is already expressed as a power: $5^{1/12}$ .
Finally, consider $\sqrt[4]{\sqrt[3]{5}}$ . This can be rewritten as $(\sqrt[3]{5})^{1/4} = (5^{1/3})^{1/4}$ . Again, using the power rule, this is $5^{1/12}$ .

All three expressions simplify to $5^{1/12}$ . Therefore, all values are equal. The correct choice is:

All values are equal.

Final Answer

All values are equal

Key Points to Remember

Essential concepts to master this topic

Power Rule: Convert nested radicals to fractional exponents first
Technique: $\sqrt[6]{\sqrt{5}} = (5^{1/2})^{1/6} = 5^{1/12}$
Check: All expressions must equal $5^{1/12}$ to be equivalent ✓

Common Mistakes

Avoid these frequent errors

Comparing radical expressions without converting to common form
Don't try to compare $\sqrt[6]{\sqrt{5}}$ and $\sqrt[12]{5}$ directly = impossible to see if equal! The different radical forms hide their true relationship. Always convert all expressions to the same base with fractional exponents.

Practice Quiz

Test your knowledge with interactive questions

Solve the following exercise:

$\sqrt[5]{\sqrt[3]{5}}=$

FAQ

Everything you need to know about this question

How do I convert nested radicals to fractional exponents?

Work from the inside out! For $\sqrt[6]{\sqrt{5}}$ , first write $\sqrt{5} = 5^{1/2}$ , then $\sqrt[6]{5^{1/2}} = (5^{1/2})^{1/6}$ .

What's the power rule for exponents again?

When you have a power raised to another power, multiply the exponents: $(a^m)^n = a^{m \cdot n}$ . So $(5^{1/2})^{1/6} = 5^{1/2 \cdot 1/6} = 5^{1/12}$ .

Why can't I just estimate which radical looks bigger?

Visual estimation fails with complex radicals! $\sqrt[6]{\sqrt{5}}$ and $\sqrt[12]{5}$ look different but are actually equal. Always convert to fractional exponents to compare accurately.

How do I handle a radical inside another radical?

Use the chain rule approach: Convert the innermost radical first, then work outward. For $\sqrt[4]{\sqrt[3]{5}}$ , start with $\sqrt[3]{5} = 5^{1/3}$ , then $\sqrt[4]{5^{1/3}} = 5^{1/12}$ .

What if the expressions aren't equal?

Then you'd compare the fractional exponents! Since $a^m > a^n$ when $a > 1$ and $m > n$ , the expression with the larger exponent would be greatest.

Can I use a calculator to check my work?

Absolutely! Calculate each expression's decimal value. If they're all equal (like 1.1487...), then all values are equal. This confirms your algebraic work!

🌟 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