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

Q: Why does the cube root of 64 equal 4?

Because 4 × 4 × 4 = 64! The cube root asks: what number multiplied by itself three times gives 64? Since $ 4^3 = 64 $, we have $ \sqrt[3]{64} = 4 $.

Q: What's the difference between √(x²) and just x?

$ \sqrt{x^2} = |x| $ (absolute value), not just x! This means if x = -2, then $ \sqrt{(-2)^2} = \sqrt{4} = 2 $, which is positive. The square root always gives a non-negative result.

Q: Should I work from inside out or outside in?

Always work inside out! Start with the innermost operation (cube root), then move to the next layer (division), then the outermost (square root). This follows the natural order of operations.

Q: Why isn't x = -2 listed as an answer choice?

Many textbooks focus on principal solutions or positive values when dealing with basic radical equations. Both x = 2 and x = -2 satisfy $ |x| = 2 $, but the question likely expects the positive solution.

Radical Equations with Nested Operations

Solve the following exercise:

$\sqrt{\frac{16}{\sqrt[3]{64}}}=\sqrt{x^2}$

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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 Square it in order to eliminate the root

00:29 Break down 64 into 4 cubed

00:35 A cube root cancels out a cube power

00:46 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:

$\sqrt{\frac{16}{\sqrt[3]{64}}}=\sqrt{x^2}$

Step-by-step solution

To solve the problem $\sqrt{\frac{16}{\sqrt[3]{64}}} = \sqrt{x^2}$ , we proceed step-by-step as follows:

First, we simplify $\sqrt[3]{64}$ . Since $64 = 4^3$ , it follows that $\sqrt[3]{64} = 4$ .
Next, simplify the expression $\frac{16}{\sqrt[3]{64}}$ :
$\frac{16}{4} = 4$ .
Now, take the square root of this simplified value. Thus, $\sqrt{4} = 2$ .
The equation simplifies to: $\sqrt{x^2} = 2$ . Since $\sqrt{x^2} = |x|$ , we have $|x| = 2$ .
This implies $x = 2$ or $x = -2$ .
However, choices include only positive solutions, and thus $x = 2$ .

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

Final Answer

$x=2$

Key Points to Remember

Essential concepts to master this topic

Order: Simplify innermost expressions first, then work outward
Technique: Calculate $\sqrt[3]{64} = 4$ , then $\frac{16}{4} = 4$
Check: Substitute x = 2: $\sqrt{2^2} = \sqrt{4} = 2$ ✓

Common Mistakes

Avoid these frequent errors

Ignoring absolute value in square root of x²
Don't assume $\sqrt{x^2} = x$ = positive x only! This misses negative solutions. $\sqrt{x^2} = |x|$ means both x = 2 and x = -2 are valid. Always remember that square roots of squared variables give absolute values.

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 does the cube root of 64 equal 4?

Because 4 × 4 × 4 = 64! The cube root asks: what number multiplied by itself three times gives 64? Since $4^3 = 64$ , we have $\sqrt[3]{64} = 4$ .

What's the difference between √(x²) and just x?

$\sqrt{x^2} = |x|$ (absolute value), not just x! This means if x = -2, then $\sqrt{(-2)^2} = \sqrt{4} = 2$ , which is positive. The square root always gives a non-negative result.

Should I work from inside out or outside in?

Always work inside out! Start with the innermost operation (cube root), then move to the next layer (division), then the outermost (square root). This follows the natural order of operations.

Why isn't x = -2 listed as an answer choice?

Many textbooks focus on principal solutions or positive values when dealing with basic radical equations. Both x = 2 and x = -2 satisfy $|x| = 2$ , but the question likely expects the positive solution.

How do I simplify nested radicals like this?

Break it into steps: 1) Simplify innermost radical first, 2) Perform any arithmetic operations, 3) Simplify the outer radical. Don't try to do everything at once!

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

Radical Equations with Nested Operations

❤️ 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 does the cube root of 64 equal 4?

What's the difference between √(x²) and just x?

Should I work from inside out or outside in?

Why isn't x = -2 listed as an answer choice?

How do I simplify nested radicals like this?

🌟 Unlock Your Math Potential

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