Calculate (1/3)² : Finding the Square of One-Third

Fraction Exponents with Perfect Squares

(13)2= (\frac{1}{3})^2=

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Step-by-step video solution

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00:00 Solve
00:05 Let's break down the exponent into multiplications
00:09 Make sure to multiply numerator by numerator and denominator by denominator
00:12 And this is the solution to the question

Step-by-step written solution

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1

Understand the problem

(13)2= (\frac{1}{3})^2=

2

Step-by-step solution

To solve for (13)2 \left( \frac{1}{3} \right)^2 , we follow these steps:

  • Step 1: Identify the given fraction, which is 13 \frac{1}{3} .
  • Step 2: Apply the formula for squaring a fraction: (ab)2=a2b2 \left( \frac{a}{b} \right)^2 = \frac{a^2}{b^2} .
  • Step 3: Square the numerator: 12=1 1^2 = 1 .
  • Step 4: Square the denominator: 32=9 3^2 = 9 .
  • Step 5: Form the new fraction using the squared values: 1232=19 \frac{1^2}{3^2} = \frac{1}{9} .

Therefore, the solution to (13)2 \left( \frac{1}{3} \right)^2 is 19 \frac{1}{9} .

3

Final Answer

19 \frac{1}{9}

Key Points to Remember

Essential concepts to master this topic
  • Rule: Square both numerator and denominator separately when raising fractions to powers
  • Technique: Apply (ab)2=a2b2 (\frac{a}{b})^2 = \frac{a^2}{b^2} , so (13)2=1232=19 (\frac{1}{3})^2 = \frac{1^2}{3^2} = \frac{1}{9}
  • Check: Multiply 19×19=181≠(13)2 \frac{1}{9} \times \frac{1}{9} = \frac{1}{81} \neq (\frac{1}{3})^2 , but 13×13=19 \frac{1}{3} \times \frac{1}{3} = \frac{1}{9} ✓

Common Mistakes

Avoid these frequent errors
  • Adding the exponent to numerator and denominator instead of multiplying
    Don't think (13)2=1+23+2=35 (\frac{1}{3})^2 = \frac{1+2}{3+2} = \frac{3}{5} ! This confuses exponent rules with addition. Always apply the exponent by multiplying: (13)2=13×13=1×13×3=19 (\frac{1}{3})^2 = \frac{1}{3} \times \frac{1}{3} = \frac{1 \times 1}{3 \times 3} = \frac{1}{9} .

Practice Quiz

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\( 11^2= \)

FAQ

Everything you need to know about this question

Why do I square both the top and bottom numbers separately?

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When you square a fraction, you're multiplying it by itself: 13×13 \frac{1}{3} \times \frac{1}{3} . Using fraction multiplication rules, multiply numerators together and denominators together: 1×13×3=19 \frac{1 \times 1}{3 \times 3} = \frac{1}{9} .

Is there a difference between (13)2 (\frac{1}{3})^2 and 1232 \frac{1^2}{3^2} ?

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No difference at all! Both expressions mean exactly the same thing. The second form 1232 \frac{1^2}{3^2} just shows the exponent rule more clearly.

What if the fraction wasn't already in lowest terms?

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Always simplify first if possible! For example, (26)2 (\frac{2}{6})^2 should become (13)2=19 (\frac{1}{3})^2 = \frac{1}{9} rather than 436 \frac{4}{36} .

How can I check my answer is correct?

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Think about what squaring means: (13)2 (\frac{1}{3})^2 asks "what times itself equals 13×13 \frac{1}{3} \times \frac{1}{3} ?" So multiply your answer by itself to verify!

Why is my answer smaller than the original fraction?

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When you square a proper fraction (numerator smaller than denominator), the result is always smaller! This is because 13<1 \frac{1}{3} < 1 , and multiplying by a number less than 1 makes things smaller.

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