Simplify the Square Root: Solving √(9/36) Step by Step

Q: Should I simplify the fraction before or after taking the square root?

Either way works! You can simplify $ \frac{9}{36} = \frac{1}{4} $ first, then find $ \sqrt{\frac{1}{4}} = \frac{1}{2} $, or take $ \sqrt{9} = 3 $ and $ \sqrt{36} = 6 $ to get $ \frac{3}{6} = \frac{1}{2} $.

Q: How do I know when I can use the square root quotient property?

You can always use $ \sqrt{\frac{a}{b}} = \frac{\sqrt{a}}{\sqrt{b}} $ as long as both a and b are positive. This property works for any positive numbers under the square root!

Q: Why does my calculator show 0.5 instead of 1/2?

That's correct! $ \frac{1}{2} = 0.5 $ - they're the same value in different forms. In math class, fraction form is usually preferred unless the problem asks for a decimal.

Square Root Properties with Fraction Simplification

Complete the following exercise:

$\sqrt{\frac{9}{36}}=$

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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

00:16 Break down 9 to 3 squared

00:21 Break down 36 to 6 squared

00:27 The root of any number (A) squared cancels out the square

00:30 Apply this formula to our exercise and proceed to cancel out the squares

00:38 Break down 6 into factors of 3 and 2

00:43 Reduce wherever possible

00:46 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:

$\sqrt{\frac{9}{36}}=$

Step-by-step solution

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

Step 1: Simplify the fraction within the square root, if possible.
Step 2: Apply the square root quotient property $\sqrt{\frac{a}{b}} = \frac{\sqrt{a}}{\sqrt{b}}$ .
Step 3: Calculate the square roots of the resulting numbers.
Step 4: Simplify the fraction, if necessary.

Now, let's work through each step:

Step 1: Simplify the fraction $\frac{9}{36}$ .
We notice that both 9 and 36 have a common factor of 9. So, we simplify:

$\frac{9}{36} = \frac{1}{4}$

Step 2: Apply the square root quotient property:

$\sqrt{\frac{9}{36}} = \sqrt{\frac{1}{4}}$

Now address the square root of the simplified fraction:

$\sqrt{\frac{1}{4}} = \frac{\sqrt{1}}{\sqrt{4}}$

Step 3: Calculate the square roots:

$\sqrt{1} = 1$
$\sqrt{4} = 2$

Step 4: Simplify the fraction:

$\frac{1}{2}$

Therefore, the solution to the problem is $\frac{1}{2}$ .

Final Answer

$\frac{1}{2}$

Key Points to Remember

Essential concepts to master this topic

Rule: Simplify fractions before taking square roots when possible
Technique: Use $\sqrt{\frac{a}{b}} = \frac{\sqrt{a}}{\sqrt{b}}$ to separate numerator and denominator
Check: Verify $(\frac{1}{2})^2 = \frac{1}{4} = \frac{9}{36}$ ✓

Common Mistakes

Avoid these frequent errors

Taking square root of numerator and denominator separately without simplifying first
Don't calculate $\frac{\sqrt{9}}{\sqrt{36}} = \frac{3}{6}$ and forget to simplify = $\frac{3}{6}$ instead of $\frac{1}{2}$ ! This gives the right numerical value but in unsimplified form. Always reduce fractions to lowest terms both before and after taking square roots.

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

Should I simplify the fraction before or after taking the square root?

Either way works! You can simplify $\frac{9}{36} = \frac{1}{4}$ first, then find $\sqrt{\frac{1}{4}} = \frac{1}{2}$ , or take $\sqrt{9} = 3$ and $\sqrt{36} = 6$ to get $\frac{3}{6} = \frac{1}{2}$ .

How do I know when I can use the square root quotient property?

You can always use $\sqrt{\frac{a}{b}} = \frac{\sqrt{a}}{\sqrt{b}}$ as long as both a and b are positive. This property works for any positive numbers under the square root!

What if the numbers under the square root aren't perfect squares?

No problem! Even if you can't simplify completely, you can still use the quotient property. For example, $\sqrt{\frac{2}{8}} = \frac{\sqrt{2}}{\sqrt{8}} = \frac{\sqrt{2}}{2\sqrt{2}} = \frac{1}{2}$ .

Why does my calculator show 0.5 instead of 1/2?

That's correct! $\frac{1}{2} = 0.5$ - they're the same value in different forms. In math class, fraction form is usually preferred unless the problem asks for a decimal.

Can I just memorize that 9/36 equals 1/4?

While memorizing common fraction equivalents helps, it's better to understand the process. Look for the greatest common factor (GCF): both 9 and 36 are divisible by 9, so $\frac{9}{36} = \frac{9÷9}{36÷9} = \frac{1}{4}$ .

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