Calculate the Square Root of 272¼: Mixed Number Evaluation

Q: Why can't I just take the square root of 272 and 1/4 separately?

Because square roots don't distribute over addition! $ \sqrt{a + b} \neq \sqrt{a} + \sqrt{b} $. You must first convert the mixed number to an improper fraction, then take the square root.

Q: How do I convert 272¼ to an improper fraction?

Multiply the whole number by the denominator: $ 272 \times 4 = 1088 $. Then add the numerator: $ 1088 + 1 = 1089 $. So $ 272\frac{1}{4} = \frac{1089}{4} $.

Q: How do I know that √1089 = 33?

You can check by multiplication: $ 33 \times 33 = 1089 $. If you don't recognize perfect squares, try numbers systematically: 30² = 900, 32² = 1024, 33² = 1089 ✓

Q: Why is 33/2 the same as 16½?

Divide 33 by 2: $ 33 ÷ 2 = 16 $ remainder $ 1 $. The remainder becomes the numerator over the same denominator: $ \frac{33}{2} = 16\frac{1}{2} $.

Q: Can I leave my answer as 33/2 instead of converting to mixed number?

Yes! Both $ \frac{33}{2} $ and $ 16\frac{1}{2} $ are correct. However, since the original problem used a mixed number, it's often preferred to give the answer in the same format.

Q: How can I check if my final answer is correct?

Square your answer! $ (16\frac{1}{2})^2 = (\frac{33}{2})^2 = \frac{33^2}{2^2} = \frac{1089}{4} = 272\frac{1}{4} $ ✓ This matches the original number under the square root.

Square Root Operations with Mixed Numbers

$\sqrt{272\frac{1}{4}}=$

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Step-by-step video solution

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00:00 Solve

00:06 Let's break down the number 16.5 squared

00:09 The square root of any number (X) squared, root cancels square

00:12 And this is the solution to the question

Step-by-step written solution

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Understand the problem

$\sqrt{272\frac{1}{4}}=$

Step-by-step solution

To solve for the square root of $272\frac{1}{4}$ , we can follow these steps:

Step 1: Convert the mixed number to an improper fraction. $272\frac{1}{4}$ becomes $\frac{1089}{4}$ .
Step 2: Find the square root of $\frac{1089}{4}$ .

Step 1: Convert the mixed number to an improper fraction:
To convert $272\frac{1}{4}$ to an improper fraction:

Multiply the whole number 272 by 4: $272 \times 4 = 1088$
Add the numerator of the fraction: $1088 + 1 = 1089$
Thus, $272\frac{1}{4}$ is $\frac{1089}{4}$ .

Step 2: Calculate the square root of $\frac{1089}{4}$ :
The square root of a fraction $\frac{a}{b}$ is $\frac{\sqrt{a}}{\sqrt{b}}$ .

$\sqrt{\frac{1089}{4}} = \frac{\sqrt{1089}}{\sqrt{4}}$
The square root of 1089 is 33, since $33 \times 33 = 1089$ .
The square root of 4 is 2, since $2 \times 2 = 4$ .

Therefore, $\sqrt{\frac{1089}{4}} = \frac{33}{2} = 16.5 = 16\frac{1}{2}$ .

Thus, the square root of $272\frac{1}{4}$ is $16\frac{1}{2}$ .

Final Answer

$16\frac{1}{2}$

Key Points to Remember

Essential concepts to master this topic

Rule: Convert mixed numbers to improper fractions before taking square root
Technique: Use $\sqrt{\frac{a}{b}} = \frac{\sqrt{a}}{\sqrt{b}}$ , so $\sqrt{\frac{1089}{4}} = \frac{33}{2}$
Check: Verify by squaring: $(16\frac{1}{2})^2 = (\frac{33}{2})^2 = \frac{1089}{4} = 272\frac{1}{4}$ ✓

Common Mistakes

Avoid these frequent errors

Taking square root of whole number and fraction separately
Don't calculate $\sqrt{272}$ and $\sqrt{\frac{1}{4}}$ separately = approximately 16.49 and 0.5! This gives a completely wrong answer because you can't split square roots across addition. Always convert to improper fraction first: $272\frac{1}{4} = \frac{1089}{4}$ .

Practice Quiz

Test your knowledge with interactive questions

$\sqrt{36}=$

FAQ

Everything you need to know about this question

Why can't I just take the square root of 272 and 1/4 separately?

Because square roots don't distribute over addition! $\sqrt{a + b} \neq \sqrt{a} + \sqrt{b}$ . You must first convert the mixed number to an improper fraction, then take the square root.

How do I convert 272¼ to an improper fraction?

Multiply the whole number by the denominator: $272 \times 4 = 1088$ . Then add the numerator: $1088 + 1 = 1089$ . So $272\frac{1}{4} = \frac{1089}{4}$ .

How do I know that √1089 = 33?

You can check by multiplication: $33 \times 33 = 1089$ . If you don't recognize perfect squares, try numbers systematically: 30² = 900, 32² = 1024, 33² = 1089 ✓

Why is 33/2 the same as 16½?

Divide 33 by 2: $33 ÷ 2 = 16$ remainder $1$ . The remainder becomes the numerator over the same denominator: $\frac{33}{2} = 16\frac{1}{2}$ .

Can I leave my answer as 33/2 instead of converting to mixed number?

Yes! Both $\frac{33}{2}$ and $16\frac{1}{2}$ are correct. However, since the original problem used a mixed number, it's often preferred to give the answer in the same format.

How can I check if my final answer is correct?

Square your answer! $(16\frac{1}{2})^2 = (\frac{33}{2})^2 = \frac{33^2}{2^2} = \frac{1089}{4} = 272\frac{1}{4}$ ✓ This matches the original number under the square root.

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