Solve the Square Root Expression: Simplifying √(196x²/49)

Q: Do I need to simplify the fraction before taking the square root?

Yes, absolutely! Simplifying $ \frac{196}{49} = 4 $ first makes the square root much easier. You get $ \sqrt{4x^2} $ instead of dealing with $ \frac{\sqrt{196x^2}}{\sqrt{49}} $.

Q: Why is the answer 2x and not just 2?

The expression under the square root is $ 4x^2 $, not just 4. When you take $ \sqrt{4x^2} $, you get $ \sqrt{4} \times \sqrt{x^2} = 2 \times x = 2x $. Don't forget the variable!

Q: What if x is negative? Does √(x²) still equal x?

Great question! Technically, $ \sqrt{x^2} = |x| $ (absolute value). But in most algebra problems like this, we assume $ x \geq 0 $ unless told otherwise, so $ \sqrt{x^2} = x $.

Q: How do I know if 196÷49 equals 4?

You can check this division: $ 49 \times 4 = 196 $. Or notice that $ 196 = 14^2 $ and $ 49 = 7^2 $, so $ \frac{196}{49} = \frac{14^2}{7^2} = \left(\frac{14}{7}\right)^2 = 2^2 = 4 $.

Q: Can I use the quotient rule for square roots?

Yes! The quotient rule says $ \sqrt{\frac{a}{b}} = \frac{\sqrt{a}}{\sqrt{b}} $. So you could solve it as $ \frac{\sqrt{196x^2}}{\sqrt{49}} = \frac{14x}{7} = 2x $. Both methods work!

Square Root Simplification with Fraction Reduction

Solve the following exercise:

$\sqrt{\frac{196x^2}{49}}=$

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:09 Let's simplify this problem step by step.

00:13 When you have the square root of a fraction, like A divided by B, think of it as two separate roots.

00:20 It's the square root of A, divided by the square root of B.

00:24 Now, we will use this approach in our exercise.

00:28 First, break down the number one ninety-six into fourteen squared.

00:35 Next, break down forty-nine into seven squared.

00:42 Remember, the square root of any number, like A squared, removes the square.

00:48 Apply this to our exercise and cancel out the squares. You'll see it gets simpler!

00:58 And there you go, that's how we solve this problem!

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

Solve the following exercise:

$\sqrt{\frac{196x^2}{49}}=$

Step-by-step solution

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

Step 1: Simplify the fraction $\frac{196x^2}{49}$ .
Step 2: Apply the square root quotient property.
Step 3: Simplify each term to find the final simplified form.

Now, let's perform each step:

Step 1: Simplify the fraction inside the square root:

$\frac{196x^2}{49} = \frac{196}{49} \times x^2$

Simplify $\frac{196}{49}$ : Since $196 \div 49 = 4$ , we have $\frac{196}{49} = 4$ .

The expression therefore simplifies to $4x^2$ .

Step 2: Apply the square root quotient property:

$\sqrt{4x^2} = \sqrt{4} \times \sqrt{x^2}$

Step 3: Simplify each term:

$\sqrt{4} = 2$ and $\sqrt{x^2} = x$ (assuming $x \geq 0$ ).

Therefore, the expression simplifies to $2x$ .

Thus, the solution to the problem is $\boxed{2x}$ .

Final Answer

$2x$

Key Points to Remember

Essential concepts to master this topic

Rule: Apply square root to numerator and denominator separately
Technique: Simplify fractions first: $\frac{196}{49} = 4$ before taking square root
Check: Verify by squaring result: $(2x)^2 = 4x^2 = \frac{196x^2}{49}$ ✓

Common Mistakes

Avoid these frequent errors

Taking square root of fraction without simplifying first
Don't jump to $\sqrt{\frac{196x^2}{49}}$ directly = harder calculation and possible errors! Students often leave fractions unsimplified making the square root much more difficult. Always simplify the fraction first to get $\sqrt{4x^2}$ , then take 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

Do I need to simplify the fraction before taking the square root?

Yes, absolutely! Simplifying $\frac{196}{49} = 4$ first makes the square root much easier. You get $\sqrt{4x^2}$ instead of dealing with $\frac{\sqrt{196x^2}}{\sqrt{49}}$ .

Why is the answer 2x and not just 2?

The expression under the square root is $4x^2$ , not just 4. When you take $\sqrt{4x^2}$ , you get $\sqrt{4} \times \sqrt{x^2} = 2 \times x = 2x$ . Don't forget the variable!

What if x is negative? Does √(x²) still equal x?

Great question! Technically, $\sqrt{x^2} = |x|$ (absolute value). But in most algebra problems like this, we assume $x \geq 0$ unless told otherwise, so $\sqrt{x^2} = x$ .

How do I know if 196÷49 equals 4?

You can check this division: $49 \times 4 = 196$ . Or notice that $196 = 14^2$ and $49 = 7^2$ , so $\frac{196}{49} = \frac{14^2}{7^2} = \left(\frac{14}{7}\right)^2 = 2^2 = 4$ .

Can I use the quotient rule for square roots?

Yes! The quotient rule says $\sqrt{\frac{a}{b}} = \frac{\sqrt{a}}{\sqrt{b}}$ . So you could solve it as $\frac{\sqrt{196x^2}}{\sqrt{49}} = \frac{14x}{7} = 2x$ . Both methods work!

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