Calculate (2/3)²: Evaluating the Square of a Fraction

Q: Why do I square both the top and bottom numbers?

Because exponents distribute over fractions! When you square $ \frac{2}{3} $, you're really multiplying $ \frac{2}{3} \times \frac{2}{3} $, which gives $ \frac{2 \times 2}{3 \times 3} = \frac{4}{9} $.

Q: What if I accidentally only squared the numerator?

You'd get $ \frac{4}{3} $ instead of $ \frac{4}{9} $! This is a very common mistake. Remember: the exponent applies to the entire fraction, so both parts get squared.

Q: How can I check if my answer is right?

Multiply your answer by itself! If $ \frac{4}{9} $ is correct, then $ \frac{4}{9} \times \frac{4}{9} = \frac{16}{81} $. You can also convert to decimals: $ \left(\frac{2}{3}\right)^2 = (0.667)^2 ≈ 0.444 = \frac{4}{9} $.

Q: Why isn't the answer just 4/6?

$ \frac{4}{6} $ comes from incorrectly thinking $ 3^2 = 6 $. But 3 squared is 9, not 6! Always double-check your basic multiplication: $ 3 \times 3 = 9 $.

Q: Do all fraction exponents work the same way?

Yes! Whether it's $ \left(\frac{1}{4}\right)^3 $ or $ \left(\frac{5}{7}\right)^4 $, you always apply the exponent to both numerator and denominator separately.

Fraction Exponents with Squaring Operations

Insert the corresponding expression:

$\left(\frac{2}{3}\right)^2=$

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Simplify the following problem

00:03 According to the power laws, a fraction raised to a power (N)

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

00:11 We will apply this formula to our exercise

00:17 Let's calculate each power and substitute accordingly

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:

$\left(\frac{2}{3}\right)^2=$

Step-by-step solution

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

Apply the exponentiation rule to the fraction $\frac{2}{3}$ .
Calculate $2^2$ and $3^2$ .
Form the result as a fraction.
Compare the result to the provided choices and select the correct one.

Now, let's work through each step:

Step 1: The expression given is $\left(\frac{2}{3}\right)^2$ .

Step 2: According to the exponentiation rule, we apply the exponent to both the numerator and the denominator:
$\left(\frac{2}{3}\right)^2 = \frac{2^2}{3^2}$ .

Step 3: Calculate $2^2$ and $3^2$ :
$2^2 = 4$ , and $3^2 = 9$ .

Step 4: Form the resultant fraction:
Thus, $\frac{2^2}{3^2} = \frac{4}{9}$ .

Step 5: Finally, compare this result with the given choices:
Our result $\frac{4}{9}$ matches with choice 2.

Therefore, the solution to the problem is $\frac{4}{9}$ .

Final Answer

$\frac{4}{9}$

Key Points to Remember

Essential concepts to master this topic

Rule: Apply exponent to both numerator and denominator separately
Technique: $\left(\frac{2}{3}\right)^2 = \frac{2^2}{3^2} = \frac{4}{9}$
Check: Verify $\frac{4}{9} \times \frac{4}{9} = \frac{16}{81}$ for powers ✓

Common Mistakes

Avoid these frequent errors

Adding exponent to numerator and denominator instead of multiplying
Don't think $\left(\frac{2}{3}\right)^2 = \frac{2+2}{3+2} = \frac{4}{5}$ ! Exponents mean repeated multiplication, not addition. Always apply the exponent by multiplying: $\left(\frac{2}{3}\right)^2 = \frac{2 \times 2}{3 \times 3} = \frac{4}{9}$ .

Practice Quiz

Test your knowledge with interactive questions

$(3\times4\times5)^4=$

FAQ

Everything you need to know about this question

Why do I square both the top and bottom numbers?

Because exponents distribute over fractions! When you square $\frac{2}{3}$ , you're really multiplying $\frac{2}{3} \times \frac{2}{3}$ , which gives $\frac{2 \times 2}{3 \times 3} = \frac{4}{9}$ .

What if I accidentally only squared the numerator?

You'd get $\frac{4}{3}$ instead of $\frac{4}{9}$ ! This is a very common mistake. Remember: the exponent applies to the entire fraction, so both parts get squared.

How can I check if my answer is right?

Multiply your answer by itself! If $\frac{4}{9}$ is correct, then $\frac{4}{9} \times \frac{4}{9} = \frac{16}{81}$ . You can also convert to decimals: $\left(\frac{2}{3}\right)^2 = (0.667)^2 ≈ 0.444 = \frac{4}{9}$ .

Why isn't the answer just 4/6?

$\frac{4}{6}$ comes from incorrectly thinking $3^2 = 6$ . But 3 squared is 9, not 6! Always double-check your basic multiplication: $3 \times 3 = 9$ .

Do all fraction exponents work the same way?

Yes! Whether it's $\left(\frac{1}{4}\right)^3$ or $\left(\frac{5}{7}\right)^4$ , you always apply the exponent to both numerator and denominator separately.

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