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Exponential Algebra
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Exponential notation and algebraic laws for powers, including a0=1, negative exponents, multiplying and dividing powers with the same base, exponents of products and quotients and powers of powers (am)n=amn.
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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🚫
Exponential Equations (Equating Indices)
AHL AI 1.9
If two exponentials in the same positive base are equal, their exponents must be equal:
an=am⇔n=m,a>0,a=1🚫
Exponentials can also appear in equations with one or more unknown:
(21)x−1=8x+1
⇒(2−1)x−1=(23)x+1
⇒21−x=23x+3
Now we can equate the exponents:
1−x=3x+3⇒x=−21
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