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Question
Prove for all x in R [(-x)^3 = -(x^3)]
(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.

Rachel, my bad! Its the associative property that you want to use, not the distributive.

Associative:  (a･b)･c = a･(b･c)  as before

Distributive:  a･(b+c) = a･b + a･c.

Sorry. Hope I didn't confuse you too much.

Randy

It looks like you just need to the commutative property for multiplication of real numbers, which means that the order of multilication does not matter, i.e,

a･b = b･a.

as well as the definition of raising a number to an integer power

x^n = x･x･ ... ･x    n times

and the distributive property

(a･b)･c = a･(b･c)

Then

(-x)^3 = [(-1)･x]^3 = [(-1)･x]･[(-1)･x]･[(-1)･x] by definition

= (-1)･[(-1)･(-1)]･(x･x･x)  commutivity and distributivity

= (-1)･x^3  by definiton and the fact that (-1)･(-1) = 1

= -(x^3).

The only reason this is tricky is that we are so used to assuming the validity of the simple manipulations above for real numbers. For more complicated objects, they don't necessarily hold (eg., commutivity for matrices).

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#### randy patton

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