# Don't Mix Up $\,3x\,$ versus $\,x^3\,$!

Some people confuse the shorthands for
repeated addition and repeated multiplication.
The purpose of this section is to give you plenty of practice,
so *you* won't confuse the two!

*Exponents* give a shorthand for
repeated multiplication.
For example:

That is,
$\,(\text{blah})^3\,$ represents three *factors* of blah.
Here are some examples:

$x^3 = x\cdot x\cdot x$

$(2x)^3 = (2x)(2x)(2x) = 8x^3$

You get the same result using an exponent law:

$\,(2x)^3 = 2^3x^3 = 8x^3\,$

$(x+2)^3 = (x+2)(x+2)(x+2)$

*Multiplication by an integer*
gives a shorthand for repeated addition.
For example:

Here are some examples:

$3x = x + x + x$

$\displaystyle \cssId{s18}{ \begin{align} &3(x+1)\cr &\ \ = (x+1) + (x+1) + (x+1)\cr &\ \ = 3x + 3 \end{align}} $

You get the same result using the distributive law:

$\,3(x+1) = 3x + 3$

$3(2x) = 2x + 2x + 2x = 6x$

You get the same result using the associative law:

$\,3(2x) = (3\cdot 2)x = 6x$