audio read-through 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:

$$\cssId{s5}{(\text{blah})^3 = \text{blah}\cdot\text{blah}\cdot\text{blah}}$$

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:

$$\cssId{s15}{\,3(\text{blah}) = \text{blah} + \text{blah} +\text{blah}}$$

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:

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

Concept Practice

What is this a shorthand for?