Solve Nested Tenth Roots: Simplifying ∛(∛1)

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

Because 1 raised to any power equals 1! Since $ \sqrt[n]{x} = x^{1/n} $, we get $ \sqrt[n]{1} = 1^{1/n} = 1 $. This is true for any positive root index.

Q: How do I combine nested roots in general?

Use the formula: $ \sqrt[n]{\sqrt[m]{x}} = \sqrt[n \cdot m]{x} $. Multiply the indices and keep the same radicand (number inside). For example: $ \sqrt[3]{\sqrt[4]{8}} = \sqrt[12]{8} $.

Q: Are all the answer choices really equivalent?

Yes! All three expressions equal 1: $ 1 = 1 $, $ \sqrt[100]{1} = 1 $, and $ \sqrt{1} = 1 $. They're different forms of the same value.

Q: What if the number inside wasn't 1?

The nested root rule still works! For example: $ \sqrt[2]{\sqrt[3]{8}} = \sqrt[6]{8} $. But you'd need to simplify further since 8 has other roots besides 1.

Q: Can I work from the inside out instead?

You can, but it's less efficient. Working inside-out: $ \sqrt[10]{1} = 1 $, then $ \sqrt[10]{1} = 1 $. Using the formula directly is faster and shows the mathematical relationship better.

Nested Radical Expressions with Equivalent Forms

Solve the following exercise:

$\sqrt[10]{\sqrt[10]{1}}=$

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Solve the following problem

00:06 When we have a number (A) in a root of order (B) in a root of order (C)

00:09 The result equals the number (A) in a root of the order of their product (B times C)

00:12 Let's apply this formula to our exercise

00:18 Calculate the order of the product

00:27 A root of any order of the number 1 always equals 1

00:36 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:

$\sqrt[10]{\sqrt[10]{1}}=$

Step-by-step solution

To solve this problem, we'll observe the following process:

Step 1: Recognize the expression $\sqrt[10]{\sqrt[10]{1}}$ involves nested roots.
Step 2: Apply the formula for nested roots: $\sqrt[n]{\sqrt[m]{x}} = \sqrt[n \cdot m]{x}$ .
Step 3: Set $n = 10$ and $m = 10$ , resulting in $\sqrt[10 \times 10]{1} = \sqrt[100]{1}$ .
Step 4: Simplify $\sqrt[100]{1}$ . Any root of 1 is 1, as $1^k = 1$ for any positive rational number $k$ .

Thus, the evaluation of the original expression $\sqrt[10]{\sqrt[10]{1}}$ equals 1.

Comparing this result to the provided choices:

Choice 1 is $1$ .
Choice 2 is $\sqrt[100]{1}$ , which is also 1.
Choice 3 is $\sqrt{1} = 1$ .
Choice 4 states all answers are correct.

Therefore, choice 4 is correct: All answers are equivalent to the solution, being 1.

Thus, the correct selection is: All answers are correct.

Final Answer

All answers are correct.

Key Points to Remember

Essential concepts to master this topic

Rule: Nested roots combine by multiplying indices: $\sqrt[n]{\sqrt[m]{x}} = \sqrt[nm]{x}$
Technique: $\sqrt[10]{\sqrt[10]{1}} = \sqrt[10 \times 10]{1} = \sqrt[100]{1}$
Check: Any root of 1 equals 1 because $1^{1/n} = 1$ for all positive n ✓

Common Mistakes

Avoid these frequent errors

Treating nested roots as separate calculations
Don't solve $\sqrt[10]{1} = 1$ first, then $\sqrt[10]{1} = 1$ = wrong process! This wastes time and misses the pattern. Always use the nested root formula to combine indices directly: $\sqrt[n]{\sqrt[m]{x}} = \sqrt[nm]{x}$ .

Practice Quiz

Test your knowledge with interactive questions

Solve the following exercise:

$\sqrt{\sqrt{4}}=$

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 $\sqrt[n]{x} = x^{1/n}$ , we get $\sqrt[n]{1} = 1^{1/n} = 1$ . This is true for any positive root index.

How do I combine nested roots in general?

Use the formula: $\sqrt[n]{\sqrt[m]{x}} = \sqrt[n \cdot m]{x}$ . Multiply the indices and keep the same radicand (number inside). For example: $\sqrt[3]{\sqrt[4]{8}} = \sqrt[12]{8}$ .

Are all the answer choices really equivalent?

Yes! All three expressions equal 1: $1 = 1$ , $\sqrt[100]{1} = 1$ , and $\sqrt{1} = 1$ . They're different forms of the same value.

What if the number inside wasn't 1?

The nested root rule still works! For example: $\sqrt[2]{\sqrt[3]{8}} = \sqrt[6]{8}$ . But you'd need to simplify further since 8 has other roots besides 1.

Can I work from the inside out instead?

You can, but it's less efficient. Working inside-out: $\sqrt[10]{1} = 1$ , then $\sqrt[10]{1} = 1$ . Using the formula directly is faster and shows the mathematical relationship better.

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