Simplify the Square Root: √(196/49) Step-by-Step Solution

Q: How do I remember which numbers are perfect squares?

Practice the squares from 1 to 15: 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, 196, 225. Notice that 196 = 14² and 49 = 7²!

Q: What if the numerator or denominator isn't a perfect square?

You can still use the quotient property! For example: $ \sqrt{\frac{18}{2}} = \frac{\sqrt{18}}{\sqrt{2}} = \frac{3\sqrt{2}}{\sqrt{2}} = 3 $. Break down any non-perfect squares into simpler radicals.

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

Yes! If $ \frac{196}{49} $ simplifies to $ \frac{4 \times 49}{49} = 4 $, then $ \sqrt{4} = 2 $. Both methods give the same answer!

Q: How do I check my answer is correct?

Square your answer and compare: $ 2^2 = 4 $, and $ \frac{196}{49} = \frac{14^2}{7^2} = \left(\frac{14}{7}\right)^2 = 2^2 = 4 $ ✓

Square Root Simplification with Fractional Radicands

Complete the following exercise:

$\sqrt{\frac{196}{49}}=$

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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 The root of the fraction (A divided by B)

00:07 Equals the root of the numerator (A) divided by the root of the denominator (B)

00:11 Apply this formula to our exercise

00:19 Factorize 196 into 14 squared

00:24 Factorize 19 into 7 squared

00:29 The root of any number (A) squared cancels out the square

00:32 Apply this formula to our exercise and proceed to cancel out the squares

00:41 This is the solution

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

Complete the following exercise:

$\sqrt{\frac{196}{49}}=$

Step-by-step solution

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

Step 1: Apply the square root quotient property.
Step 2: Calculate the square roots of the numerator and the denominator.
Step 3: Simplify the resulting expression.

Now, let's work through each step:
Step 1: Apply the square root quotient property $\sqrt{\frac{196}{49}} = \frac{\sqrt{196}}{\sqrt{49}}$ .
Step 2: Calculate the individual square roots: $\sqrt{196} = 14$ and $\sqrt{49} = 7$ .
Step 3: Simplify the expression to $\frac{14}{7} = 2$ .

Therefore, the solution to the problem is 2.

Final Answer

Key Points to Remember

Essential concepts to master this topic

Quotient Property: $\sqrt{\frac{a}{b}} = \frac{\sqrt{a}}{\sqrt{b}}$ separates numerator and denominator
Perfect Squares: Identify $\sqrt{196} = 14$ and $\sqrt{49} = 7$
Verify: Check that $2^2 = 4$ and $\frac{196}{49} = 4$ ✓

Common Mistakes

Avoid these frequent errors

Taking square root of fraction as a whole
Don't calculate $\sqrt{196/49}$ by first dividing 196 ÷ 49 = 4, then $\sqrt{4} = 2$ ! While this gives the right answer here, it creates confusion and won't work with non-perfect square quotients. Always use the quotient property to separate the square roots first.

Practice Quiz

Test your knowledge with interactive questions

Choose the expression that is equal to the following:

$\sqrt{a}:\sqrt{b}$

FAQ

Everything you need to know about this question

Why can't I just divide 196 by 49 first, then take the square root?

While this works when the quotient is a perfect square (like 4), it won't work for problems like $\sqrt{\frac{50}{8}}$ . Using the quotient property is the reliable method that works every time!

How do I remember which numbers are perfect squares?

Practice the squares from 1 to 15: 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, 196, 225. Notice that 196 = 14² and 49 = 7²!

What if the numerator or denominator isn't a perfect square?

You can still use the quotient property! For example: $\sqrt{\frac{18}{2}} = \frac{\sqrt{18}}{\sqrt{2}} = \frac{3\sqrt{2}}{\sqrt{2}} = 3$ . Break down any non-perfect squares into simpler radicals.

Can I simplify the fraction before taking the square root?

Yes! If $\frac{196}{49}$ simplifies to $\frac{4 \times 49}{49} = 4$ , then $\sqrt{4} = 2$ . Both methods give the same answer!

How do I check my answer is correct?

Square your answer and compare: $2^2 = 4$ , and $\frac{196}{49} = \frac{14^2}{7^2} = \left(\frac{14}{7}\right)^2 = 2^2 = 4$ ✓

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