Help with Proving (-x)3=-(x3) in Real Numbers

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In summary, the conversation is about proving the statement (-x)3 = -(x3) for all real numbers x. It is mentioned that the fact -x = (-1)*x can be used, but the focus should be on using axioms. The speaker suggests writing (-x)3 as (-1)(x)(x)(x) and -(x)3 as (-1)((x)(x)(x)) to show their equivalence. However, they are unsure if this constitutes a sufficient proof. The other speaker points out that (-x)3 can also be written as (-x)(-x)(-x), which can then be shown to be equal to -(x)(x)(x).
  • #1

Homework Statement

Prove for all xεR [(-x)3 = -(x3)]
(Hint you may use the fact that -x = (-1)*x, but other wise stick to axioms)
I wrote something along the lines of .
(-x)3 can be written as (-1)(x)(x)(x) and -(x)3 can be written as (-1)((x)(x)(x))
and these are both equivalent
But it doesn't feel like I'm proving anything.
Not sure how to really write this out so there is enough "proof"
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  • #2
I think the point you are missing is that (-x)3 "can be written as" (-x)(-x)(-x) and then you can show that is equal to -(x)(x)(x).

1. What does (-x)3 mean?

(-x)3 means the quantity of negative x cubed. This means that you multiply -x by itself three times.

2. How do you prove (-x)3 = -(x3) in real numbers?

To prove this equation, you can use the basic properties of real numbers. Start by expanding (-x)3 to (-x)(-x)(-x) and then use the commutative and associative properties to rearrange the terms. Finally, use the definition of negative numbers to show that -(x3) is equal to (-x)(-x)(-x).

3. Can you provide an example to help understand (-x)3 = -(x3)?

Sure, let's use the number -2 for x. (-(-2))^3 is equal to (-2)(-2)(-2) which is equal to -8. On the other hand, -(-2^3) is equal to -(8) which is also equal to -8. This shows that (-x)3 is equal to -(x3) in this case.

4. Why is it important to prove (-x)3 = -(x3) in real numbers?

This proof is important because it demonstrates a fundamental property of real numbers. It shows that when you raise a negative number to an odd power, the result will also be negative. This knowledge is useful in solving more complex equations and understanding the behavior of negative numbers in mathematical operations.

5. Can this equation be proven using other methods?

Yes, there are multiple ways to prove this equation. One method is to use the properties of exponents and the definition of negative numbers, while another method is to use the distributive property and the definition of negative numbers. Both methods will lead to the same conclusion that (-x)3 is equal to -(x3).

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