Compare Cube Values: 3³ vs (-3)³ Mathematical Analysis

Cube Calculations with Negative Signs

Which is larger?

33□−(−3)3 3^3\square-(-3)^3

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations
00:00 Put the appropriate sign
00:04 First let's calculate the sign
00:07 Odd power, therefore the sign remains negative
00:19 Negative times negative always equals positive
00:24 We can see that the numbers are equal
00:27 And this is the solution to the question

Step-by-step written solution

Follow each step carefully to understand the complete solution
1

Understand the problem

Which is larger?

33□−(−3)3 3^3\square-(-3)^3

2

Step-by-step solution

To compare 333^3 and −(−3)3-(-3)^3, we will calculate each expression separately and then determine their relationship:

  • Calculate 333^3:

    333^3 means 3×3×33 \times 3 \times 3.

    First, 3×3=93 \times 3 = 9.

    Then, 9×3=279 \times 3 = 27.

    Therefore, 33=273^3 = 27.

  • Calculate −(−3)3-(-3)^3:

    (−3)3(-3)^3 means (−3)×(−3)×(−3)(-3) \times (-3) \times (-3).

    First, (−3)×(−3)=9(-3) \times (-3) = 9. (two negatives multiply to a positive)

    Then, 9×(−3)=−279 \times (-3) = -27. (positive times negative is negative)

    Thus, (−3)3=−27(-3)^3 = -27.

    Considering the negative sign: −(−3)3=−(−27)=27-(-3)^3 = -(-27) = 27.

After calculating, we find that both expressions equal 27. Thus, they are equal.

The correct choice is:

= =

3

Final Answer

= =

Key Points to Remember

Essential concepts to master this topic
  • Rule: Calculate exponents before applying negative signs outside parentheses
  • Technique: (−3)3=−27(-3)^3 = -27, then −(−27)=27-(-27) = 27
  • Check: Both 333^3 and −(−3)3-(-3)^3 equal 27 ✓

Common Mistakes

Avoid these frequent errors
  • Confusing negative base with negative sign placement
    Don't treat −(−3)3-(-3)^3 as −(33)-(3^3) = wrong negative result! The parentheses around -3 mean the negative is part of the base being cubed. Always calculate (−3)3(-3)^3 first, then apply the outer negative sign.

Practice Quiz

Test your knowledge with interactive questions

\( (-2)^7= \)

FAQ

Everything you need to know about this question

Why does cubing a negative number give a negative result?

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When you multiply an odd number of negative values, the result is negative. Since (−3)3=(−3)×(−3)×(−3)(-3)^3 = (-3) \times (-3) \times (-3), you're multiplying three negatives together.

What's the difference between −33-3^3 and (−3)3(-3)^3?

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−33=−(33)=−27-3^3 = -(3^3) = -27 (negative applied after cubing), while (−3)3=(−3)×(−3)×(−3)=−27(-3)^3 = (-3) \times (-3) \times (-3) = -27 (negative is part of the base). Parentheses matter!

How do I handle the double negative in −(−3)3-(-3)^3?

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Work from the inside out: First calculate (−3)3=−27(-3)^3 = -27, then apply the outer negative: −(−27)=+27-(-27) = +27. Two negatives make a positive!

Is there a pattern for odd vs even exponents with negatives?

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Yes! When the base is negative:

  • Odd exponents keep the negative sign
  • Even exponents become positive
This is because you multiply an odd or even number of negative signs.

How can I quickly check if my comparison is correct?

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Calculate each side separately and compare the final numbers: 33=273^3 = 27 and −(−3)3=27-(-3)^3 = 27. Since 27 = 27, they're equal!

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