Solve the Cube Expression: Simplifying 1/8³ Step by Step

Negative Exponents with Fraction Conversion

183=? \frac{1}{8^3}=\text{?}

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations
00:03 Let's learn how to switch a fraction with a positive exponent into one with a negative exponent.
00:09 When a number A is raised to the power of negative N, it's equal to one divided by A raised to the power of N.
00:16 So, if we use this rule, we can change numbers between fractions and whole numbers easily.
00:22 For example, one divided by eight becomes eight.
00:26 And the power of three changes to negative three.
00:30 And that's the solution!

Step-by-step written solution

Follow each step carefully to understand the complete solution
1

Understand the problem

183=? \frac{1}{8^3}=\text{?}

2

Step-by-step solution

We use the negative exponent rule.

b−n=1bn b^{-n}=\frac{1}{b^n}

We apply it to the problem in the opposite sense.:

183=8−3 \frac{1}{8^3}=8^{-3}

Therefore, the correct answer is option A.

3

Final Answer

8−3 8^{-3}

Key Points to Remember

Essential concepts to master this topic
  • Rule: 1bn=b−n \frac{1}{b^n} = b^{-n} converts fractions to negative exponents
  • Technique: Apply rule backwards: 183=8−3 \frac{1}{8^3} = 8^{-3}
  • Check: Verify 8−3=183=1512 8^{-3} = \frac{1}{8^3} = \frac{1}{512} ✓

Common Mistakes

Avoid these frequent errors
  • Confusing negative exponents with negative numbers
    Don't write 183=−83 \frac{1}{8^3} = -8^3 = -512! The negative sign applies to the exponent position, not the number itself. Always remember 8−3 8^{-3} means reciprocal of 83 8^3 , which is positive.

Practice Quiz

Test your knowledge with interactive questions

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

FAQ

Everything you need to know about this question

What does a negative exponent really mean?

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A negative exponent means "take the reciprocal and flip the sign." So 8−3 8^{-3} means "1 divided by 8 cubed" or 183 \frac{1}{8^3} .

Why isn't the answer −83 -8^3 ?

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Because −83=−512 -8^3 = -512 (negative number), but 183=1512 \frac{1}{8^3} = \frac{1}{512} (positive fraction). The negative exponent rule doesn't make numbers negative!

How do I remember which direction the rule goes?

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Think: "Fraction form ↔ Negative exponent form". You can go both ways! 1bn=b−n \frac{1}{b^n} = b^{-n} and b−n=1bn b^{-n} = \frac{1}{b^n} .

What if I need to calculate the actual decimal value?

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First find 83=8×8×8=512 8^3 = 8 \times 8 \times 8 = 512 , then 1512=0.001953125 \frac{1}{512} = 0.001953125 . But for this problem, 8−3 8^{-3} is the correct simplified form!

Are there other ways to write this answer?

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Yes! 183=8−3=1512=0.001953125 \frac{1}{8^3} = 8^{-3} = \frac{1}{512} = 0.001953125 are all equivalent. But 8−3 8^{-3} is the most simplified exponential form.

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