# Question a7e02

Nov 29, 2016

Actually, Josiah is correct

#### Explanation:

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

If $x = 0$

1) ${\left(- 2 x\right)}^{3} = {\left(- 2 \cdot 0\right)}^{3} = {0}^{3} = 0$

2) $- 2 {x}^{3} = - 2 \cdot {0}^{3} = - 2 \cdot 0 = 0$

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

If $x = 1$

1) ${\left(- 2 \cdot 1\right)}^{3} = {\left(- 2\right)}^{3} = - 8$

2) $- 2 \cdot {1}^{3} = - 2 \cdot 1 = - 2$

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

${\left(- 2 x\right)}^{3} = \left(- 2 x\right) \left(- 2 x\right) \left(- 2 x\right) = \left(- 2\right) \left(- 2\right) \left(- 2\right) \left(x\right) \left(x\right) \left(x\right) = - 8 {x}^{3}$

And the second expression is

$- 2 {x}^{3}$

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

Nov 30, 2016

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

#### Explanation:

$\textcolor{b l u e}{\text{Consider: } {\left(- 2 x\right)}^{3}}$

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

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

$\left(- 2 x\right) \times \left(- 2 x\right) \times \left(- 2 x\right)$

This is the same as:

${\left(- 2\right)}^{3} \times {\left(x\right)}^{3} = - 8 {x}^{3}$

So $\text{ } \textcolor{b l u e}{{\left(- 2 x\right)}^{3} = - 8 {x}^{3}}$

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

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:

$\left(- 2\right) \times {x}^{3} \text{ "=" } - 2 {x}^{3}$

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
$\textcolor{b l u e}{\text{Comparing the two}}$

It is stated that ${\left(- 2 x\right)}^{3} = - 2 {x}^{3}$

$\implies - 8 {x}^{3} = - 2 {x}^{3}$

$\textcolor{b r o w n}{\text{Consider the case "x!=0}}$

Divide both sies by ${x}^{3}$ giving

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

$\text{So for "x!=0:" } - 8 {x}^{3} \ne - 2 {x}^{3}$
..................................................................................................

$\textcolor{b r o w n}{\text{Consider the case "x=0}}$

For $x = 0 : \text{ } - 8 {x}^{3} = 02 {x}^{3}$

$- 8 {\left(0\right)}^{3} = - 2 {\left(0\right)}^{2}$

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

$\text{So for "x=0:" } - 8 {x}^{3} = - 2 {x}^{3}$