Solve Nested Roots: Finding Fifth Root of Square Root of 1024

Q: Can I just multiply the root indices together?

Yes! For nested radicals, $ \sqrt[a]{\sqrt[b]{x}} = \sqrt[ab]{x} $. So $ \sqrt[5]{\sqrt{1024}} = \sqrt[5 \times 2]{1024} = \sqrt[10]{1024} $.

Q: How do I recognize that 1024 is a perfect power?

Look for patterns! $ 1024 = 2^{10} $ because $ 2^{10} = 1024 $. Powers of 2 are common: 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024.

Q: What if I can't see the perfect power pattern?

Use prime factorization! Break down 1024 into prime factors: $ 1024 = 2 \times 2 \times 2... $ until you get $ 2^{10} $.

Q: Can I use a calculator for this?

Yes, but understanding the exponential properties helps you solve faster! $ 1024^{\frac{1}{10}} $ on a calculator gives 2, confirming our algebraic work.

Nested Radicals with Exponential Properties

Solve the following exercise:

$\sqrt[5]{\sqrt[]{1024}}=$

❤️ 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

00:06 Let's solve this math problem together.

00:09 A regular root is called a square root. It has an order of two.

00:14 If we have a number, A, under two roots: one of order B, and another of order C.

00:22 The answer is A, under a root of order B times C.

00:27 Let's use this rule for our exercise.

00:32 Now, if A is to the power of B, under a root order C.

00:43 The answer is A to the power of B divided by C.

00:47 We'll use this formula for our problem.

00:51 And there you have it. That's our solution!

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

Solve the following exercise:

$\sqrt[5]{\sqrt[]{1024}}=$

Step-by-step solution

To solve this problem, we'll follow these steps:

Step 1: Simplify the inner square root.
Step 2: Apply the formula for nested roots.
Step 3: Compute the fifth root.

Now, let's work through each step:

Step 1: Calculate the inner square root.
We have $\sqrt{1024}$ . We know that $1024 = 2^{10}$ , so $\sqrt{1024} = \sqrt{2^{10}}$ .

Applying the property of roots, $\sqrt{2^{10}} = 2^{\frac{10}{2}} = 2^5 = 32$ .

Step 2: Now, apply the fifth root to the result from step 1.
We need to find $\sqrt[5]{32}$ .

Step 3: Simplify using the properties of exponents.
From $\sqrt[5]{32} = \sqrt[5]{2^5}$ , we have $2^{\frac{5}{5}} = 2^1 = 2$ .

Therefore, the solution to the problem is $2$ .

Final Answer

Key Points to Remember

Essential concepts to master this topic

Rule: Simplify inner root first, then apply outer root
Technique: Convert to powers: $\sqrt[5]{\sqrt{1024}} = (1024^{\frac{1}{2}})^{\frac{1}{5}} = 1024^{\frac{1}{10}}$
Check: Verify $2^{10} = 1024$ , so $1024^{\frac{1}{10}} = 2$ ✓

Common Mistakes

Avoid these frequent errors

Trying to combine roots incorrectly
Don't add the root indices like $\sqrt[5+2]{1024} = \sqrt[7]{1024}$ = wrong answer! This ignores the order of operations and gives incorrect results. Always work from inside out: calculate the inner root first, then apply the outer root.

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

Can I just multiply the root indices together?

Yes! For nested radicals, $\sqrt[a]{\sqrt[b]{x}} = \sqrt[ab]{x}$ . So $\sqrt[5]{\sqrt{1024}} = \sqrt[5 \times 2]{1024} = \sqrt[10]{1024}$ .

How do I recognize that 1024 is a perfect power?

Look for patterns! $1024 = 2^{10}$ because $2^{10} = 1024$ . Powers of 2 are common: 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024.

What if I can't see the perfect power pattern?

Use prime factorization! Break down 1024 into prime factors: $1024 = 2 \times 2 \times 2...$ until you get $2^{10}$ .

Why does the order matter in nested radicals?

Just like parentheses in arithmetic, you must work from the innermost operation outward. Calculate $\sqrt{1024} = 32$ first, then find $\sqrt[5]{32}$ .

Can I use a calculator for this?

Yes, but understanding the exponential properties helps you solve faster! $1024^{\frac{1}{10}}$ on a calculator gives 2, confirming our 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