Calculate (2/3)³: Finding the Cube of Two-Thirds

Fraction Exponents with Cubing Operations

(23)3= (\frac{2}{3})^3=

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Step-by-step written solution

Follow each step carefully to understand the complete solution
1

Understand the problem

(23)3= (\frac{2}{3})^3=

2

Step-by-step solution

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

  • Step 1: Identify the given fraction, which is 23\frac{2}{3}.
  • Step 2: Apply the formula for exponents applied to fractions, (ab)n=anbn \displaystyle(\frac{a}{b})^n = \frac{a^n}{b^n} .
  • Step 3: Calculate the cube of the numerator and the cube of the denominator separately.
  • Step 4: Write the results as a single fraction.

Now, let's work through each step:

Step 1: The problem provides the fraction 23\frac{2}{3}.

Step 2: Use the formula (ab)n=anbn(\frac{a}{b})^n = \frac{a^n}{b^n} for a=2a = 2, b=3b = 3, and n=3n = 3.

Step 3: We need to calculate 232^3 and 333^3:
- Calculate 23=2×2×2=82^3 = 2 \times 2 \times 2 = 8.
- Calculate 33=3×3×3=273^3 = 3 \times 3 \times 3 = 27.

Step 4: Writing these as a fraction gives 2333=827 \displaystyle \frac{2^3}{3^3} = \frac{8}{27} .

Therefore, the solution to the problem is 827\frac{8}{27}.

3

Final Answer

827 \frac{8}{27}

Key Points to Remember

Essential concepts to master this topic
  • Rule: When raising a fraction to a power, raise numerator and denominator separately
  • Technique: Apply (ab)n=anbn (\frac{a}{b})^n = \frac{a^n}{b^n} to get 2333=827 \frac{2^3}{3^3} = \frac{8}{27}
  • Check: Multiply 827×827×827 \frac{8}{27} \times \frac{8}{27} \times \frac{8}{27} should equal our answer ✓

Common Mistakes

Avoid these frequent errors
  • Only cubing the numerator and forgetting the denominator
    Don't calculate (23)3 (\frac{2}{3})^3 as 233=83 \frac{2^3}{3} = \frac{8}{3} ! This ignores the exponent rule and gives a completely wrong result. Always raise both the numerator AND denominator to the given power.

Practice Quiz

Test your knowledge with interactive questions

Which of the following is equivalent to the expression below?

\( \)\( 10,000^1 \)

FAQ

Everything you need to know about this question

Why do I need to cube both the top and bottom numbers?

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Because when you raise a fraction to a power, the entire fraction is being multiplied by itself that many times. So (23)3=23×23×23 (\frac{2}{3})^3 = \frac{2}{3} \times \frac{2}{3} \times \frac{2}{3} , which means both 2 and 3 get multiplied three times each.

What's the difference between 233 \frac{2^3}{3} and (23)3 (\frac{2}{3})^3 ?

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Great question! 233=83 \frac{2^3}{3} = \frac{8}{3} means only the numerator is cubed. But (23)3=2333=827 (\frac{2}{3})^3 = \frac{2^3}{3^3} = \frac{8}{27} means the entire fraction is cubed, so both parts get the exponent.

How do I remember which number goes where?

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Think of it as "what's on top stays on top, what's on bottom stays on bottom". The 2 (numerator) becomes 23=8 2^3 = 8 on top, and the 3 (denominator) becomes 33=27 3^3 = 27 on bottom.

Can I simplify 827 \frac{8}{27} any further?

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No, 827 \frac{8}{27} is already in simplest form! Since 8 = 2³ and 27 = 3³, they share no common factors other than 1, so this fraction cannot be reduced.

What if the original fraction could be simplified first?

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It doesn't matter! You can either simplify first then cube, or cube first then simplify. You'll get the same answer both ways. But since 23 \frac{2}{3} is already simplified, we just cube it directly.

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