Simplify the Square Root Expression: √(64x²/x⁴)

Q: Why do I get a negative exponent when dividing?

When the exponent in the denominator is larger than the numerator, you get negative exponents! $ x^2 ÷ x^4 = x^{2-4} = x^{-2} $. This is normal and means one over that variable.

Q: How do I handle the square root of a negative exponent?

$ \sqrt{x^{-2}} = x^{-1} = \frac{1}{x} $ because half of -2 equals -1. Remember: $ \sqrt{x^n} = x^{n/2} $ for any exponent!

Q: Can I simplify the fraction after taking the square root?

It's much easier to simplify first, then take the square root! Reducing $ \frac{64x^2}{x^4} $ to $ 64x^{-2} $ makes the square root step cleaner.

Q: What if x could be negative?

For this problem, we assume x is positive since we're dealing with real square roots. If x were negative, we'd need to consider absolute values: $ \sqrt{x^2} = |x| $.

Q: Is there a faster way to solve this?

Yes! You can use the property $ \sqrt{\frac{a}{b}} = \frac{\sqrt{a}}{\sqrt{b}} $ directly: $ \sqrt{\frac{64x^2}{x^4}} = \frac{\sqrt{64x^2}}{\sqrt{x^4}} = \frac{8x}{x^2} = \frac{8}{x} $

Radical Simplification with Exponential Division

Solve the following exercise:

$\sqrt{\frac{64x^2}{x^4}}=$

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Simplify the following problem

00:03 When there's a root of a fraction (A divided by B)

00:06 It can be written as root of numerator (A) divided by root of denominator (B)

00:09 We'll apply this formula to our exercise

00:15 When there's a root of multiplication (A times B)

00:19 It can be divided into root of (A) times root of (B)

00:22 We'll apply this formula to our exercise

00:32 Let's factor 64 into 8 squared

00:38 Let's factor X to the fourth power into X squared squared

00:44 The root of any number (A) squared cancels out the square

00:47 Apply this formula to our exercise, and proceed to cancel out the squares

00:59 Let's factor X squared into factors X and X

01:05 Reduce wherever possible

01:09 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:

$\sqrt{\frac{64x^2}{x^4}}=$

Step-by-step solution

To solve this problem, we'll simplify the expression inside the square root step-by-step:

Step 1: Simplify the fraction $\frac{64x^2}{x^4}$ using the quotient rule for exponents:

$\frac{64x^2}{x^4} = 64 \cdot x^{2-4} = 64 \cdot x^{-2}$ .

Step 2: Now apply the square root to the simplified expression:

$\sqrt{\frac{64x^2}{x^4}} = \sqrt{64x^{-2}} = \sqrt{64} \cdot \sqrt{x^{-2}}$ .
$\sqrt{64} = 8$ and $\sqrt{x^{-2}} = x^{-1}$ (since it represents inverse squaring).
Thus, $\sqrt{64x^{-2}} = 8 \cdot x^{-1} = \frac{8}{x}$ .

Therefore, the solution to the problem is $\frac{8}{x}$ , which matches choice 1.

Final Answer

$\frac{8}{x}$

Key Points to Remember

Essential concepts to master this topic

Exponent Rule: When dividing powers, subtract exponents: $x^2 ÷ x^4 = x^{-2}$
Technique: Simplify inside radical first: $\sqrt{64x^{-2}} = \sqrt{64} \cdot \sqrt{x^{-2}} = 8x^{-1}$
Check: Substitute back: $\sqrt{\frac{64x^2}{x^4}} = \sqrt{\frac{64}{x^2}} = \frac{8}{x}$ ✓

Common Mistakes

Avoid these frequent errors

Applying square root to numerator and denominator separately
Don't write $\sqrt{\frac{64x^2}{x^4}} = \frac{\sqrt{64x^2}}{\sqrt{x^4}}$ = $\frac{8x}{x^2}$ without simplifying first! This skips the crucial step of reducing the fraction inside the radical. Always simplify the fraction completely before applying the 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 do I get a negative exponent when dividing?

When the exponent in the denominator is larger than the numerator, you get negative exponents! $x^2 ÷ x^4 = x^{2-4} = x^{-2}$ . This is normal and means one over that variable.

How do I handle the square root of a negative exponent?

$\sqrt{x^{-2}} = x^{-1} = \frac{1}{x}$ because half of -2 equals -1. Remember: $\sqrt{x^n} = x^{n/2}$ for any exponent!

Can I simplify the fraction after taking the square root?

It's much easier to simplify first, then take the square root! Reducing $\frac{64x^2}{x^4}$ to $64x^{-2}$ makes the square root step cleaner.

What if x could be negative?

For this problem, we assume x is positive since we're dealing with real square roots. If x were negative, we'd need to consider absolute values: $\sqrt{x^2} = |x|$ .

Is there a faster way to solve this?

Yes! You can use the property $\sqrt{\frac{a}{b}} = \frac{\sqrt{a}}{\sqrt{b}}$ directly: $\sqrt{\frac{64x^2}{x^4}} = \frac{\sqrt{64x^2}}{\sqrt{x^4}} = \frac{8x}{x^2} = \frac{8}{x}$

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