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

Q: Why do I need to apply the exponent to both parts of the fraction?

When you raise a fraction to a power, you're multiplying the entire fraction by itself. So $ \left(\frac{2}{3}\right)^2 = \frac{2}{3} \times \frac{2}{3} = \frac{2 \times 2}{3 \times 3} = \frac{2^2}{3^2} $.

Q: What if the exponent is bigger than 2?

The same rule applies! For example, $ \left(\frac{2}{3}\right)^3 = \frac{2^3}{3^3} = \frac{8}{27} $. Just apply the exponent to both the numerator and denominator.

Q: Can I use this rule with negative fractions?

Yes! For $ \left(-\frac{2}{3}\right)^2 $, you get $ \frac{(-2)^2}{3^2} = \frac{4}{9} $. Remember that negative times negative equals positive!

Fraction Exponents with Power Rules

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 laws of exponents, a fraction raised the 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

00:14 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 will apply the exponent rule for fractions:

Step 1: We are given the expression $\left(\frac{2}{3}\right)^2$ .
Step 2: Apply the fraction exponent rule: $\left(\frac{a}{b}\right)^n = \frac{a^n}{b^n}$ .

Applying this rule to our expression:

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

Calculating further would give:

$\frac{2^2}{3^2} = \frac{4}{9}$ .

However, the question asks to only match the expression, which is $\frac{2^2}{3^2}$ .

The correct choice from the given options is $\frac{2^2}{3^2}$ .

This matches Choice 3 in the provided multiple choices.

Final Answer

$\frac{2^2}{3^2}$

Key Points to Remember

Essential concepts to master this topic

Power Rule: Apply exponent to both numerator and denominator separately
Technique: $\left(\frac{2}{3}\right)^2 = \frac{2^2}{3^2}$ keeps exponents visible
Check: Final calculation gives $\frac{4}{9}$ when simplified ✓

Common Mistakes

Avoid these frequent errors

Only applying the exponent to numerator or denominator
Don't write $\left(\frac{2}{3}\right)^2 = \frac{2^2}{3}$ or $\frac{2}{3^2}$ = wrong fraction! The exponent affects the entire fraction. Always apply the power to both the top AND bottom: $\left(\frac{a}{b}\right)^n = \frac{a^n}{b^n}$ .

Practice Quiz

Test your knowledge with interactive questions

$112^0=\text{?}$

FAQ

Everything you need to know about this question

Why do I need to apply the exponent to both parts of the fraction?

When you raise a fraction to a power, you're multiplying the entire fraction by itself. So $\left(\frac{2}{3}\right)^2 = \frac{2}{3} \times \frac{2}{3} = \frac{2 \times 2}{3 \times 3} = \frac{2^2}{3^2}$ .

Should I calculate the final answer or leave it with exponents?

It depends on what the question asks for! This problem wanted the expression form $\frac{2^2}{3^2}$ , but you could also simplify to $\frac{4}{9}$ if needed.

What if the exponent is bigger than 2?

The same rule applies! For example, $\left(\frac{2}{3}\right)^3 = \frac{2^3}{3^3} = \frac{8}{27}$ . Just apply the exponent to both the numerator and denominator.

Can I use this rule with negative fractions?

Yes! For $\left(-\frac{2}{3}\right)^2$ , you get $\frac{(-2)^2}{3^2} = \frac{4}{9}$ . Remember that negative times negative equals positive!

What about fractional exponents?

That's a more advanced topic involving roots! For now, focus on whole number exponents like 2, 3, 4, etc. The same rule applies: distribute the exponent to both parts.

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