Evaluate (1/2)²: Computing the Square of One-Half

Q: Why can't I just multiply the fraction by itself?

You absolutely can multiply the fraction by itself! $ \frac{1}{2} \times \frac{1}{2} = \frac{1 \times 1}{2 \times 2} = \frac{1}{4} $. This gives the same result as using the power rule.

Q: What's the difference between (1/2)² and 1/2²?

Big difference! $ (\frac{1}{2})^2 = \frac{1}{4} $ squares the entire fraction, but $ \frac{1}{2^2} = \frac{1}{4} $ only squares the denominator. Parentheses matter!

Q: How do I remember the power rule for fractions?

Think: "Whatever I do to the whole, I do to each part." If you're squaring the fraction, square both the top number and the bottom number separately.

Q: Is 1/4 the same as 2/8 or other equivalent fractions?

Yes! $ \frac{1}{4} = \frac{2}{8} = \frac{3}{12} $ are all equivalent. But always give your answer in simplest form unless told otherwise.

Q: Why is the answer smaller than the original fraction?

When you square a proper fraction (less than 1), the result is always smaller! $ \frac{1}{2} = 0.5 $ and $ (\frac{1}{2})^2 = \frac{1}{4} = 0.25 $. This makes sense!

Fraction Exponents with Squaring Operations

Insert 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

00:00 Simplify the following problem

00:04 According to the laws of exponents, a fraction raised to a power (N)

00:08 equals the numerator and denominator, each raised to the same power (N)

00:11 We will apply this formula to our exercise

00:16 Let's calculate each power and substitute accordingly , 1 raised to any power is always equal to 1

00:25 This is the solution

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

Insert the corresponding expression:

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

Step-by-step solution

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

Step 1: Identify the given fraction and power: $\frac{1}{2}$ and the power $2$ .
Step 2: Apply the formula $(\frac{a}{b})^n = \frac{a^n}{b^n}$ .
Step 3: Calculate the numerator as $1^2 = 1$ .
Step 4: Calculate the denominator as $2^2 = 4$ .
Step 5: Combine the results to find $\left(\frac{1}{2}\right)^2 = \frac{1}{4}$ .

Now, let's work through each step in detail:
Step 1: The fraction given is $\frac{1}{2}$ and the power is 2.
Step 2: Applying the formula $(\frac{a}{b})^n = \frac{a^n}{b^n}$ , we have:
Step 3: Calculate $1^2 = 1$ .
Step 4: Calculate $2^2 = 4$ .
Step 5: Compile these to get $\frac{1}{4}$ .

Therefore, the solution to the problem is $\frac{1}{4}$ , which matches choice 2.

Final Answer

$\frac{1}{4}$

Key Points to Remember

Essential concepts to master this topic

Rule: When raising fractions to powers, raise both numerator and denominator
Technique: Apply $(\frac{a}{b})^n = \frac{a^n}{b^n}$ so $(\frac{1}{2})^2 = \frac{1^2}{2^2}$
Check: Verify $\frac{1}{4} \times \frac{1}{4} = \frac{1}{4}$ by multiplying $\frac{1}{2} \times \frac{1}{2}$ ✓

Common Mistakes

Avoid these frequent errors

Only squaring the numerator or denominator
Don't square just the top (1² = 1) or just the bottom (2² = 4) separately = incomplete calculation! This leads to wrong fractions like $\frac{1}{2}$ or $\frac{1}{4}$ without proper reasoning. Always square both the numerator AND denominator using the power rule for fractions.

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 the fraction by itself?

You absolutely can multiply the fraction by itself! $\frac{1}{2} \times \frac{1}{2} = \frac{1 \times 1}{2 \times 2} = \frac{1}{4}$ . This gives the same result as using the power rule.

What's the difference between (1/2)² and 1/2²?

Big difference! $(\frac{1}{2})^2 = \frac{1}{4}$ squares the entire fraction, but $\frac{1}{2^2} = \frac{1}{4}$ only squares the denominator. Parentheses matter!

How do I remember the power rule for fractions?

Think: "Whatever I do to the whole, I do to each part." If you're squaring the fraction, square both the top number and the bottom number separately.

Is 1/4 the same as 2/8 or other equivalent fractions?

Yes! $\frac{1}{4} = \frac{2}{8} = \frac{3}{12}$ are all equivalent. But always give your answer in simplest form unless told otherwise.

Why is the answer smaller than the original fraction?

When you square a proper fraction (less than 1), the result is always smaller! $\frac{1}{2} = 0.5$ and $(\frac{1}{2})^2 = \frac{1}{4} = 0.25$ . This makes sense!

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