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

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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

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\( 11^2= \)

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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