Calculate (1/3)³: Evaluating the Cube of a Simple Fraction

Exponent Rules with Simple Fractions

Insert the corresponding expression:

(13)3= \left(\frac{1}{3}\right)^3=

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1

Understand the problem

Insert the corresponding expression:

(13)3= \left(\frac{1}{3}\right)^3=

2

Step-by-step solution

To solve the expression (13)3 \left(\frac{1}{3}\right)^3 , we need to apply the rule for exponents of a fraction, which states:

(ab)n=anbn \left(\frac{a}{b}\right)^n = \frac{a^n}{b^n}

Using this property, we can rewrite the fraction with its exponent as follows:

(13)3=1333 \left(\frac{1}{3}\right)^3 = \frac{1^3}{3^3}

Now, calculate the powers of the numerator and the denominator separately:

  • 13=1 1^3 = 1

  • 33=27 3^3 = 27

Thus, putting it all together, we have:

1333=127 \frac{1^3}{3^3} = \frac{1}{27}

This shows that raising both the numerator and the denominator of a fraction to a power involves calculating the power of each part separately and then constructing a new fraction.

The solution to the question is: 127 \frac{1}{27}

3

Final Answer

127 \frac{1}{27}

Key Points to Remember

Essential concepts to master this topic
  • Rule: Apply exponent to both numerator and denominator separately
  • Technique: (13)3=1333=127 \left(\frac{1}{3}\right)^3 = \frac{1^3}{3^3} = \frac{1}{27}
  • Check: Verify 127×127×127=127 \frac{1}{27} \times \frac{1}{27} \times \frac{1}{27} = \frac{1}{27} ✓

Common Mistakes

Avoid these frequent errors
  • Only applying the exponent to the denominator
    Don't calculate (13)3 \left(\frac{1}{3}\right)^3 as 133=127 \frac{1}{3^3} = \frac{1}{27} by accident! This happens to give the right answer here but fails for other fractions. Always apply the exponent rule: (ab)n=anbn \left(\frac{a}{b}\right)^n = \frac{a^n}{b^n} to both parts.

Practice Quiz

Test your knowledge with interactive questions

\( 112^0=\text{?} \)

FAQ

Everything you need to know about this question

Why does 13 1^3 still equal 1?

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Any power of 1 always equals 1! This is because 1 × 1 × 1 = 1. So 13=110=1100=1 1^3 = 1^{10} = 1^{100} = 1 - no matter what the exponent is.

How do I calculate 33 3^3 without a calculator?

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33 3^3 means 3 × 3 × 3. First: 3 × 3 = 9. Then: 9 × 3 = 27. So 33=27 3^3 = 27 .

What if the fraction was (25)3 \left(\frac{2}{5}\right)^3 ?

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Use the same rule! (25)3=2353=8125 \left(\frac{2}{5}\right)^3 = \frac{2^3}{5^3} = \frac{8}{125} . Calculate each part separately: 23=8 2^3 = 8 and 53=125 5^3 = 125 .

Can I just multiply the fraction by itself three times?

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Yes, that works too! 13×13×13=1×1×13×3×3=127 \frac{1}{3} \times \frac{1}{3} \times \frac{1}{3} = \frac{1 \times 1 \times 1}{3 \times 3 \times 3} = \frac{1}{27} . Both methods give the same answer!

Why is the answer smaller than the original fraction?

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When you raise a proper fraction (less than 1) to a power, it gets smaller! Think about it: 13 \frac{1}{3} of 13 \frac{1}{3} of 13 \frac{1}{3} is a tiny piece.

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