Simplify Square Root Expression: √(64x⁴/16x²)

Q: Why do I need to simplify inside the square root first?

Simplifying first makes the problem much easier! It's simpler to find $ \sqrt{4x^2} $ than to work with the complex fraction $ \sqrt{\frac{64x^4}{16x^2}} $ directly.

Q: How do I divide the exponents correctly?

Use the quotient rule: $ \frac{x^a}{x^b} = x^{a-b} $. So $ \frac{x^4}{x^2} = x^{4-2} = x^2 $. Remember to subtract the bottom exponent from the top!

Q: What if x could be negative?

Great question! When dealing with $ \sqrt{x^2} $, the result is always |x| (absolute value). However, in algebra problems like this, we typically assume variables represent positive values unless stated otherwise.

Q: Can I use a calculator for this problem?

While calculators help check your work, it's important to understand the steps. Practice simplifying by hand first: divide 64÷16=4, then use exponent rules for the variables.

Q: Why is the answer 2x and not 4x or 8x?

The key is $ \sqrt{4x^2} = \sqrt{4} \times \sqrt{x^2} = 2 \times x = 2x $. Remember: $ \sqrt{4} = 2 $, not 4! The square root of 4 is 2, not the number itself.

Square Root Simplification with Fractional Expressions

Solve the following exercise:

$\sqrt{\frac{64x^4}{16x^2}}=$

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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 is a root of a fraction (A divided by B)

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

00:09 We will apply this formula to our exercise

00:16 When there is a root of a product (A times B)

00:20 It can be divided into root of (A) times root of (B)

00:24 Apply this formula to our exercise

00:32 Factor 64 into 8 squared

00:36 Factor X to the fourth power into X squared squared

00:42 Factor 16 into 4 squared

00:50 The root of any number (A) squared cancels out the square

00:56 We will apply this formula to our exercise, and proceed to cancel out the squares

01:16 Let's factor X squared into factors X and X

01:19 Simplify wherever possible

01:22 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:

$\sqrt{\frac{64x^4}{16x^2}}=$

Step-by-step solution

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

Step 1: Simplify the expression inside the square root.
Step 2: Apply the square root property to the simplified expression.
Step 3: Confirm the result with the answer choices.

Now, let's work through each step:

Step 1: Simplify the expression inside the square root:
We have $\frac{64x^4}{16x^2}$ . Divide the coefficients and the powers of $x$ :
$\frac{64}{16} = 4$ and using exponents, $\frac{x^4}{x^2} = x^{4-2} = x^2$ .
Therefore, $\frac{64x^4}{16x^2} = 4x^2$ .

Step 2: Apply the square root property:
$\sqrt{4x^2} = \sqrt{4} \cdot \sqrt{x^2} = 2 \cdot x = 2x$ .

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

Final Answer

$2x$

Key Points to Remember

Essential concepts to master this topic

Division Rule: Simplify inside the square root first before applying
Technique: Divide coefficients and subtract exponents: $\frac{64x^4}{16x^2} = 4x^2$
Check: Verify $\sqrt{4x^2} = 2x$ by squaring: $(2x)^2 = 4x^2$ ✓

Common Mistakes

Avoid these frequent errors

Taking square root of numerator and denominator separately
Don't write $\frac{\sqrt{64x^4}}{\sqrt{16x^2}}$ = $\frac{8x^2}{4x} = 2x$ ! This works here by luck, but fails with many expressions. Always simplify the fraction inside the square root first, then take the square root of the simplified result.

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 need to simplify inside the square root first?

Simplifying first makes the problem much easier! It's simpler to find $\sqrt{4x^2}$ than to work with the complex fraction $\sqrt{\frac{64x^4}{16x^2}}$ directly.

How do I divide the exponents correctly?

Use the quotient rule: $\frac{x^a}{x^b} = x^{a-b}$ . So $\frac{x^4}{x^2} = x^{4-2} = x^2$ . Remember to subtract the bottom exponent from the top!

What if x could be negative?

Great question! When dealing with $\sqrt{x^2}$ , the result is always |x| (absolute value). However, in algebra problems like this, we typically assume variables represent positive values unless stated otherwise.

Can I use a calculator for this problem?

While calculators help check your work, it's important to understand the steps. Practice simplifying by hand first: divide 64÷16=4, then use exponent rules for the variables.

Why is the answer 2x and not 4x or 8x?

The key is $\sqrt{4x^2} = \sqrt{4} \times \sqrt{x^2} = 2 \times x = 2x$ . Remember: $\sqrt{4} = 2$ , not 4! The square root of 4 is 2, not the number itself.

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