Number Comparison Exercise: Identifying the Largest Value

Q: Why does any root of 1 always equal 1?

Because 1 raised to any power equals 1! Since $ \sqrt[n]{1} $ asks "what number to the nth power gives 1?", the answer is always 1.

Q: Do I need to calculate the nested roots step by step?

Yes, work from the inside out! For $ \sqrt[10]{\sqrt[3]{1}} $, first find $ \sqrt[3]{1} = 1 $, then $ \sqrt[10]{1} = 1 $.

Q: Could the answer be different if the number wasn't 1?

Absolutely! For example, $ \sqrt[10]{\sqrt[3]{8}} $ would give different results. With 8: $ \sqrt[3]{8} = 2 $, then $ \sqrt[10]{2} \approx 1.07 $.

Q: What if the question asked for the smallest value instead?

The answer would still be "All answers are correct" because all expressions equal 1. Whether you're looking for largest, smallest, or any specific value, they're all the same!

Radical Expressions with Multiple Nested Roots

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 The root of any order of the number 1 is always equal to 1

00:09 Apply the same method to all the expressions and determine the largest value

00:19 They are all equal, and that's 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'll simplify each expression involving the roots of 1:

Simplify $\sqrt[10]{\sqrt[10]{1}}$ :
Since $\sqrt[10]{1} = 1$ , then $\sqrt[10]{\sqrt[10]{1}} = \sqrt[10]{1} = 1$ .
Simplify $\sqrt[10]{\sqrt[3]{1}}$ :
Since $\sqrt[3]{1} = 1$ , then $\sqrt[10]{\sqrt[3]{1}} = \sqrt[10]{1} = 1$ .
Simplify $\sqrt[5]{\sqrt{1}}$ :
Since $\sqrt{1} = 1$ , then $\sqrt[5]{\sqrt{1}} = \sqrt[5]{1} = 1$ .

Upon simplifying, each of the options results in the value 1. Therefore, all expressions are equal.

The correct answer is: "All answers are correct".

Final Answer

All answers are correct

Key Points to Remember

Essential concepts to master this topic

Rule: Any root of 1 equals 1 regardless of index
Technique: Simplify from inside out: $\sqrt[3]{1} = 1$ , then $\sqrt[10]{1} = 1$
Check: All expressions simplify to 1, so comparison shows equality ✓

Common Mistakes

Avoid these frequent errors

Thinking different root indices create different values
Don't assume $\sqrt[10]{1}$ differs from $\sqrt[5]{1}$ = confusion about which is larger! Any positive number raised to any power to equal 1 must be 1 itself. Always remember that $\sqrt[n]{1} = 1$ for any positive integer n.

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

Why does any root of 1 always equal 1?

Because 1 raised to any power equals 1! Since $\sqrt[n]{1}$ asks "what number to the nth power gives 1?", the answer is always 1.

Do I need to calculate the nested roots step by step?

Yes, work from the inside out! For $\sqrt[10]{\sqrt[3]{1}}$ , first find $\sqrt[3]{1} = 1$ , then $\sqrt[10]{1} = 1$ .

Could the answer be different if the number wasn't 1?

Absolutely! For example, $\sqrt[10]{\sqrt[3]{8}}$ would give different results. With 8: $\sqrt[3]{8} = 2$ , then $\sqrt[10]{2} \approx 1.07$ .

How can I be sure all the expressions are equal?

Calculate each one completely! Every expression with nested roots of 1 will simplify to 1, making them all equal. When all values are the same, "All answers are correct" is the right choice.

What if the question asked for the smallest value instead?

The answer would still be "All answers are correct" because all expressions equal 1. Whether you're looking for largest, smallest, or any specific value, they're all the same!

🌟 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