Simplify the Nested Radical: √(16/∛64) Step-by-Step Solution

Q: Why can't I just take the square root of 16 and the cube root separately?

Because the entire fraction is under the square root! You must simplify $ \frac{16}{\sqrt[3]{64}} $ first to get one number, then take its square root.

Q: How do I remember that $ \sqrt[3]{64} = 4 $ ?

Think of perfect cubes: $ 1^3 = 1 $, $ 2^3 = 8 $, $ 3^3 = 27 $, $ 4^3 = 64 $. Practice these common cube roots to recognize them quickly!

Q: What if I get confused about which radical to solve first?

Always work from the inside out! Start with the innermost radical (cube root), then move outward. Think of it like peeling an onion - inner layers first.

Q: Can I use a calculator for cube roots?

Yes, but try to recognize perfect cubes first! For $ \sqrt[3]{64} $, knowing $ 4^3 = 64 $ is faster than using a calculator.

Q: How do I check if my final answer is correct?

Work backwards! If your answer is 2, check: $ 2^2 = 4 $, and $ \frac{16}{4} = 4 $, so $ \sqrt{\frac{16}{\sqrt[3]{64}}} = \sqrt{4} = 2 $ ✓

Nested Radicals with Cube Roots

Solve the following exercise:

$\sqrt{\frac{16}{\sqrt[3]{64}}}=$

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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 Break down 64 to 4 in third power

00:12 A cube root cancels out a power of three

00:20 Calculate the quotient

00:26 Break down 4 to 2 squared

00:29 The root cancels out the square

00:32 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:

$\sqrt{\frac{16}{\sqrt[3]{64}}}=$

Step-by-step solution

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

Step 1: Calculate the cube root of 64
Step 2: Simplify the fraction $\frac{16}{\sqrt[3]{64}}$
Step 3: Simplify $\sqrt{\text{result from step 2}}$

Let's proceed with each step:
Step 1: The cube root of 64 is calculated as follows:
$\sqrt[3]{64} = 4$ . This is because $4^3 = 64$ .

Step 2: Now, simplify the fraction $\frac{16}{4}$ :
$\frac{16}{4} = 4$ .

Step 3: Finally, take the square root of the result from step 2:
$\sqrt{4} = 2$ .

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

Final Answer

Key Points to Remember

Essential concepts to master this topic

Order: Simplify inner radical first, then work outward systematically
Technique: Calculate $\sqrt[3]{64} = 4$ since $4^3 = 64$
Check: Verify final answer: $2^2 = 4$ and $\frac{16}{4} = 4$ ✓

Common Mistakes

Avoid these frequent errors

Taking square root of numerator and denominator separately
Don't calculate $\frac{\sqrt{16}}{\sqrt{\sqrt[3]{64}}}$ = incorrect result! This violates the radical rule and creates undefined expressions. Always simplify what's inside the radical first, then take the outer square root.

Practice Quiz

Test your knowledge with interactive questions

Solve the following exercise:

$\sqrt{\frac{2}{4}}=$

FAQ

Everything you need to know about this question

Why can't I just take the square root of 16 and the cube root separately?

Because the entire fraction is under the square root! You must simplify $\frac{16}{\sqrt[3]{64}}$ first to get one number, then take its square root.

How do I remember that $\sqrt[3]{64} = 4$ ?

Think of perfect cubes: $1^3 = 1$ , $2^3 = 8$ , $3^3 = 27$ , $4^3 = 64$ . Practice these common cube roots to recognize them quickly!

What if I get confused about which radical to solve first?

Always work from the inside out! Start with the innermost radical (cube root), then move outward. Think of it like peeling an onion - inner layers first.

Can I use a calculator for cube roots?

Yes, but try to recognize perfect cubes first! For $\sqrt[3]{64}$ , knowing $4^3 = 64$ is faster than using a calculator.

How do I check if my final answer is correct?

Work backwards! If your answer is 2, check: $2^2 = 4$ , and $\frac{16}{4} = 4$ , so $\sqrt{\frac{16}{\sqrt[3]{64}}} = \sqrt{4} = 2$ ✓

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Simplify the Nested Radical: √(16/∛64) Step-by-Step Solution

Nested Radicals with Cube Roots

❤️ Continue Your Math Journey!

Step-by-step video solution

Step-by-step written solution

Understand the problem

Step-by-step solution

Final Answer

Key Points to Remember

Common Mistakes

Practice Quiz

FAQ

Why can't I just take the square root of 16 and the cube root separately?

How do I remember that $\sqrt[3]{64} = 4$ ?

What if I get confused about which radical to solve first?

Can I use a calculator for cube roots?

How do I check if my final answer is correct?

🌟 Unlock Your Math Potential

More Questions

Root of a Root

Solve the Nested Radical Equation: √(16/∛64) = √(x²)

Solve the Nested Radical Equation: √√81 = ∛√x^6

Solve √(36/144) × √√16: Multiple Square Root Operations

Multiply Nested Roots: Solving √⁷(√5) × √¹⁴(√5)

Rules of Roots Combined

Solve the Nested Root Equation: Fifth Root of Square Root of x^10 = √√81

Solve the Square Root Equation: x^6 = √16 · √25

Solve Nested Square Roots: √√(100/25) × √√25 Multiplication Problem

Solve the Expression: Cube Root of Square Root of 64 Times Square Root of 64

Simplify the Nested Radical: √(16/∛64) Step-by-Step Solution

Nested Radicals with Cube Roots

❤️ Continue Your Math Journey!

Step-by-step video solution

Step-by-step written solution

Understand the problem

Step-by-step solution

Final Answer

Key Points to Remember

Common Mistakes

Practice Quiz

FAQ

Why can't I just take the square root of 16 and the cube root separately?

How do I remember that 643=4 \sqrt[3]{64} = 4 364​=4?

What if I get confused about which radical to solve first?

Can I use a calculator for cube roots?

How do I check if my final answer is correct?

🌟 Unlock Your Math Potential

More Questions

Root of a Root

Solve the Nested Radical Equation: √(16/∛64) = √(x²)

Solve the Nested Radical Equation: √√81 = ∛√x^6

Solve √(36/144) × √√16: Multiple Square Root Operations

Multiply Nested Roots: Solving √⁷(√5) × √¹⁴(√5)

Rules of Roots Combined

Solve the Nested Root Equation: Fifth Root of Square Root of x^10 = √√81

Solve the Square Root Equation: x^6 = √16 · √25

Solve Nested Square Roots: √√(100/25) × √√25 Multiplication Problem

Solve the Expression: Cube Root of Square Root of 64 Times Square Root of 64

How do I remember that $\sqrt[3]{64} = 4$ ?