Calculate (1/2)²: Finding the Square of One-Half

Q: Why can't I just multiply 2 × 1/2 to get the square?

That's multiplication, not squaring! Squaring means multiplying by itself: $ \left(\frac{1}{2}\right)^2 = \frac{1}{2} \times \frac{1}{2} $. The expression $ 2 \times \frac{1}{2} = 1 $ is completely different.

Q: Do I need to calculate 1² and 2² separately?

Yes! Even though $ 1^2 = 1 $ seems obvious, showing both steps demonstrates proper understanding: $ \left(\frac{1}{2}\right)^2 = \frac{1^2}{2^2} = \frac{1}{4} $.

Q: What if the fraction has a bigger numerator or denominator?

The same rule applies! For example: $ \left(\frac{3}{5}\right)^2 = \frac{3^2}{5^2} = \frac{9}{25} $. Just square each part separately and simplify if needed.

Q: How do I know which answer choice is correct?

Look for the choice that applies the exponent to both parts of the fraction. Only $ \frac{1^2}{2^2} $ shows the correct exponent rule being applied.

Q: Can I convert to decimal first then square?

You could calculate $ \frac{1}{2} = 0.5 $, then $ (0.5)^2 = 0.25 $, but the question asks for the expression form, not the decimal answer.

Exponent Rules with Fractional Bases

Choose the corresponding expression:

$\left(\frac{1}{2}\right)^2=$

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:07 Let's simplify this problem together.

00:10 Remember, when a fraction is raised to a power, the exponent applies to both the top and bottom numbers.

00:18 This means both the numerator and the denominator are raised to the power of N. Let's see how that works.

00:25 We'll use this rule to solve our exercise. Ready?

00:29 And there you have it. That's how we solve the problem. Great job!

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

Choose the corresponding expression:

$\left(\frac{1}{2}\right)^2=$

Step-by-step solution

To solve this problem, we must square the fraction $\left(\frac{1}{2}\right)$ . The exponent rule for fractions states that when you raise a fraction $\left(\frac{a}{b}\right)$ to the power of $n$ , it becomes $\frac{a^n}{b^n}$ .

Here, in the fraction $\left(\frac{1}{2}\right)$ , we identify the numerator $a = 1$ and the denominator $b = 2$ , with the exponent $n = 2$ .

Applying the formula, we calculate the result:
$\left(\frac{1}{2}\right)^2 = \frac{1^2}{2^2}$

Therefore, the expression that corresponds to $\left(\frac{1}{2}\right)^2$ is $\frac{1^2}{2^2}$ , which directly matches the given choice.

The correct choice from the answer options is:

Choice 3: $\frac{1^2}{2^2}$

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

Final Answer

$\frac{1^2}{2^2}$

Key Points to Remember

Essential concepts to master this topic

Rule: When raising a fraction to a power, apply the exponent to both numerator and denominator
Technique: $\left(\frac{a}{b}\right)^n = \frac{a^n}{b^n}$ , so $\left(\frac{1}{2}\right)^2 = \frac{1^2}{2^2}$
Check: Verify that $\frac{1^2}{2^2} = \frac{1}{4}$ equals 0.25 ✓

Common Mistakes

Avoid these frequent errors

Applying the exponent to only the numerator or denominator
Don't square just the numerator to get $\frac{1^2}{2} = \frac{1}{2}$ or just the denominator! This ignores the exponent rule and gives wrong results. Always apply the exponent to both the numerator and denominator: $\left(\frac{1}{2}\right)^2 = \frac{1^2}{2^2}$ .

Practice Quiz

Test your knowledge with interactive questions

$112^0=\text{?}$

FAQ

Everything you need to know about this question

Why can't I just multiply 2 × 1/2 to get the square?

That's multiplication, not squaring! Squaring means multiplying by itself: $\left(\frac{1}{2}\right)^2 = \frac{1}{2} \times \frac{1}{2}$ . The expression $2 \times \frac{1}{2} = 1$ is completely different.

Do I need to calculate 1² and 2² separately?

Yes! Even though $1^2 = 1$ seems obvious, showing both steps demonstrates proper understanding: $\left(\frac{1}{2}\right)^2 = \frac{1^2}{2^2} = \frac{1}{4}$ .

What if the fraction has a bigger numerator or denominator?

The same rule applies! For example: $\left(\frac{3}{5}\right)^2 = \frac{3^2}{5^2} = \frac{9}{25}$ . Just square each part separately and simplify if needed.

How do I know which answer choice is correct?

Look for the choice that applies the exponent to both parts of the fraction. Only $\frac{1^2}{2^2}$ shows the correct exponent rule being applied.

Can I convert to decimal first then square?

You could calculate $\frac{1}{2} = 0.5$ , then $(0.5)^2 = 0.25$ , but the question asks for the expression form, not the decimal answer.

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