Exponential Algebra

Exponential Notation

SL Core 1.5

Exponential expressions are a shortcut for writing the multiplication of a number by itself many times:

an=a×a×⋯×an times🚫

Here a is called the base and n the exponent. We say that a is raised to the nth power.

Note that a1=a, since we have 1×a=a.

Exponent with zero

SL Core 1.5

Any number raised to the power zero is

a0=1×a×a×⋯×a0 times=1🚫

And since any number multiplied by 0 is 0:

0n=0,n=0🚫

When n=0, we have 00, which is technically undefined, but in most contexts is defined to be

00=1🚫

Multiplying powers with the same base

SL Core 1.5

When multiplying exponentials with the same base, the following rule applies:

an⋅am =a×a×⋯×an times×a×a×⋯×am times =a×a×⋯×an+m times =am+n🚫

Exponential of exponential

SL Core 1.5

An exponential can be the base of another exponential:

(am)n=a×⋯×am times×⋯×a×⋯×am timesn times=anm🚫

Negative exponents

SL Core 1.5

a−n=an1,a=0🚫

Dividing exponents with the same base

SL Core 1.5

In general,

aman=an⋅a−m=an−m,a=0🚫

Exponents of products & quotients

SL Core 1.5

When exponentials with the same power are being multiplied or divided, the bases can be combined:

anbn=(ab)n🚫

bnan=(ba)n,b=0🚫

Exercises

Key Skills

Exponential Notation

Exponent with zero

Multiplying powers with the same base

Exponential of exponential

Negative exponents

Dividing exponents with the same base

Exponents of products & quotients

Exponential Notation

Exponent with zero

Multiplying powers with the same base

Exponential of exponential

Negative exponents

Dividing exponents with the same base

Exponents of products & quotients