Question

Question #a7e02

Answer 1

Actually, Josiah is correct

Explanation:

If `x` was equal to zero then the two expressions will be equal

If `x=0`

1) `(-2x)^3=(-2*0)^3=0^3=0`

2) `-2x^3=-2*0^3=-2*0=0`

But if it was any number else then the expressions won't be equal

If `x=1`

1) `(-2*1)^3=(-2)^3=-8`

2) `-2*1^3=-2*1=-2`

This is because the first expression was raised to the power of `3`

`(-2x)^3=(-2x)(-2x)(-2x)=(-2)(-2)(-2)(x)(x)(x)=-8x^3`

And the second expression is

`-2x^3`

`-8x^3` is not the same as `-2x^3`, so they will produce different values, except for `x=0`

Answer 2

They are the same if `x=0` otherwise not the same

Explanation:

`color(blue)("Consider: "(-2x)^3)`

The brackets group together the -2 and the `x`

So the index (power) is applied to everything inside the bracket giving:

`(-2x)xx(-2x)xx(-2x)`

This is the same as:

`(-2)^3xx(x)^3 = -8x^3`

So `" "color(blue)((-2x)^3=-8x^3)`

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

`color(blue)("Consider: "-2x^3`

In this case the -2 and the `x` are not 'locked' together other than by the operation of multiply. So we have:

`(-2)xxx^3" "=" "-2x^3`

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
`color(blue)("Comparing the two")`

It is stated that `(-2x)^3=-2x^3`

`=> -8x^3=-2x^3`

`color(brown)("Consider the case "x!=0")`

Divide both sies by `x^3` giving

`-8x^3=-2x^3" "->" " -8=-2 color(red)(larr" False")`

`"So for "x!=0:" " -8x^3!=-2x^3`
..................................................................................................

`color(brown)("Consider the case "x=0")`

For `x=0:" "-8x^3=02x^3`

`-8(0)^3=-2(0)^2`

`0=0" "color(red)(larr" True")`

`"So for "x=0:" " -8x^3=-2x^3`