Solve the Nested Radical: Seventh Root of Square Root of 2

Q: Why can't I just simplify the inner radical first?

You absolutely can! $ \sqrt{2} $ is already in simplest form, but converting to rational exponents makes the next step much clearer. It helps you see exactly how to combine the operations.

Q: How do I know when to multiply exponents?

Multiply exponents when you have a power raised to another power: $ (a^m)^n = a^{m \cdot n} $. In nested radicals, the outer root applies to the entire inner expression.

Q: What's the difference between this and adding exponents?

You add exponents when multiplying like bases: $ a^m \cdot a^n = a^{m+n} $. You multiply exponents when raising a power to a power: $ (a^m)^n = a^{mn} $.

Q: Can I check my answer without a calculator?

Yes! Think logically: $ \sqrt[14]{2} $ should be closer to 1 than $ \sqrt{2} \approx 1.41 $ because we're taking a higher root. The 14th root makes the result smaller.

Q: What if I have three nested radicals?

Same principle! Convert each to rational exponents and multiply all the exponents together. For example: $ \sqrt[3]{\sqrt[4]{\sqrt{2}}} = 2^{1/2 \cdot 1/4 \cdot 1/3} = 2^{1/24} $

Nested Radicals with Rational Exponents

Solve the following exercise:

$\sqrt[7]{\sqrt{2}}=$

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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 (X) in a root of the order (B) in a root of the order (A)

00:13 The result equals the number (X) in a root of the order of their product (A times B)

00:18 Let's apply this formula to our exercise

00:25 Let's calculate the order of the product

00:31 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:

$\sqrt[7]{\sqrt{2}}=$

Step-by-step solution

Let's solve the given problem by following these steps:

Step 1: Recognize the expression $\sqrt[7]{\sqrt{2}}$ . It involves two roots.
Step 2: Rewrite each part using rational exponents. We have $\sqrt{2} = 2^{1/2}$ .
Step 3: Substitute back, giving $\sqrt[7]{2^{1/2}}$ or $(2^{1/2})^{1/7}$ .
Step 4: Use the properties of exponents: $(a^m)^n = a^{m \cdot n}$ .
Step 5: Calculate the exponent: $(1/2) \cdot (1/7) = 1/14$ .
Step 6: This gives us $2^{1/14}$ , which is equal to $\sqrt[14]{2}$ .

Thus, the simplified expression is $\sqrt[14]{2}$ .

Therefore, the solution to the problem is $\sqrt[14]{2}$ .

Final Answer

$\sqrt[14]{2}$

Key Points to Remember

Essential concepts to master this topic

Rule: Convert nested radicals to rational exponents before simplifying
Technique: Use $(a^m)^n = a^{m \cdot n}$ to get $2^{(1/2) \cdot (1/7)} = 2^{1/14}$
Check: Verify that $\sqrt[14]{2}$ raised to 14th power gives 2 ✓

Common Mistakes

Avoid these frequent errors

Adding the denominators instead of multiplying exponents
Don't add 2 + 7 = 9 to get $\sqrt[9]{2}$ = wrong answer! When dealing with nested radicals, the denominators combine through multiplication of exponents, not addition. Always multiply the exponents: (1/2) × (1/7) = 1/14.

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 simplify the inner radical first?

You absolutely can! $\sqrt{2}$ is already in simplest form, but converting to rational exponents makes the next step much clearer. It helps you see exactly how to combine the operations.

How do I know when to multiply exponents?

Multiply exponents when you have a power raised to another power: $(a^m)^n = a^{m \cdot n}$ . In nested radicals, the outer root applies to the entire inner expression.

What's the difference between this and adding exponents?

You add exponents when multiplying like bases: $a^m \cdot a^n = a^{m+n}$ . You multiply exponents when raising a power to a power: $(a^m)^n = a^{mn}$ .

Can I check my answer without a calculator?

Yes! Think logically: $\sqrt[14]{2}$ should be closer to 1 than $\sqrt{2} \approx 1.41$ because we're taking a higher root. The 14th root makes the result smaller.

What if I have three nested radicals?

Same principle! Convert each to rational exponents and multiply all the exponents together. For example: $\sqrt[3]{\sqrt[4]{\sqrt{2}}} = 2^{1/2 \cdot 1/4 \cdot 1/3} = 2^{1/24}$

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