Solve the Nested Square Root: Simplifying √(√8)

Nested Radicals with Exponent Laws

Solve the following exercise:

8= \sqrt[]{\sqrt{8}}=

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations
00:05 Let's solve this problem together.
00:08 Remember, a regular root is usually of order two.
00:13 Imagine we have a number, A, under a root of order B, then again under a root of order C.
00:20 The result is the number, A, under a root. This new root's order is the product of B and C.
00:28 Let's apply this formula to our problem now.
00:31 First, we calculate the multiplication of the orders.
00:37 If we have a number, A, raised to the power of B, under a root of order C.
00:43 Then, the result is A raised to the power of B divided by C.
00:49 Let's apply this approach to our problem again.
00:53 And there you have the solution! Great job tackling this problem!

Step-by-step written solution

Follow each step carefully to understand the complete solution
1

Understand the problem

Solve the following exercise:

8= \sqrt[]{\sqrt{8}}=

2

Step-by-step solution

In order to solve the given problem, we'll follow these steps:

  • Step 1: Convert the inner square root to an exponent: 8=81/2\sqrt{8} = 8^{1/2}.

  • Step 2: Apply the root of a root property: 8=(8)1/2=(81/2)1/2\sqrt{\sqrt{8}} = (\sqrt{8})^{1/2} = (8^{1/2})^{1/2}.

  • Step 3: Simplify the expression using exponent rules: (81/2)1/2=8(1/2)⋅(1/2)=81/4(8^{1/2})^{1/2} = 8^{(1/2) \cdot (1/2)} = 8^{1/4}.

The nested root expression simplifies to 81/48^{1/4}.

Therefore, the simplified expression of 8\sqrt{\sqrt{8}} is 814 8^{\frac{1}{4}} .

After comparing this result with the multiple choice answers, choice 2 is correct.

3

Final Answer

814 8^{\frac{1}{4}}

Key Points to Remember

Essential concepts to master this topic
  • Rule: Convert nested roots to exponents using power-of-a-power law
  • Technique: 8=(81/2)1/2=8(1/2)(1/2)=81/4 \sqrt{\sqrt{8}} = (8^{1/2})^{1/2} = 8^{(1/2)(1/2)} = 8^{1/4}
  • Check: Verify by working backwards: (81/4)4=8 (8^{1/4})^4 = 8 ✓

Common Mistakes

Avoid these frequent errors
  • Adding exponents instead of multiplying
    Don't add the exponents: 81/2+1/2=81=8 8^{1/2 + 1/2} = 8^1 = 8 ! This gives completely wrong results because you're breaking the power-of-a-power rule. Always multiply exponents when taking a power of a power: (am)n=amn (a^m)^n = a^{mn} .

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 do I need to convert to exponents?

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Converting radicals to exponents makes it easier to apply the power laws! a=a1/2 \sqrt{a} = a^{1/2} lets you use the rule (am)n=amn (a^m)^n = a^{mn} instead of memorizing separate radical rules.

What does 81/4 8^{1/4} actually mean?

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81/4 8^{1/4} means the fourth root of 8, or 84 \sqrt[4]{8} . It's the number that when raised to the 4th power gives you 8.

Can I simplify 81/4 8^{1/4} further?

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Not easily! Since 8=23 8 = 2^3 , we get 81/4=(23)1/4=23/4 8^{1/4} = (2^3)^{1/4} = 2^{3/4} , but 81/4 8^{1/4} is already the simplest form for most purposes.

How do I handle three nested roots like 8 \sqrt{\sqrt{\sqrt{8}}} ?

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Use the same method! 8=((81/2)1/2)1/2=8(1/2)(1/2)(1/2)=81/8 \sqrt{\sqrt{\sqrt{8}}} = ((8^{1/2})^{1/2})^{1/2} = 8^{(1/2)(1/2)(1/2)} = 8^{1/8} . Just keep multiplying the exponents.

Why isn't the answer just 8 \sqrt{8} ?

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Because we have two square root operations! 8 \sqrt{\sqrt{8}} means "take the square root of the result of taking the square root of 8." This is different from just 8 \sqrt{8} .

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