The Mathematics 11
Competency Test
Laws of Exponents
(i) multiplication of two powers:
multiply by 2 five times
23 x 25 = (2 x 2 x 2) x (2 x 2 x 2 x 2 x 2) = 28
multiply by 2
three times
net effect is to multiply with 2 a
total of 3 + 5 = 8 times
Thus
23 × 25 = 23+5 = 28
In symbols, we can write that if c is any number, then
c n ic m = c n +m
The illustration above which shows why multiplying two exponentials together gives a new
exponential whose exponent is the sum of the original exponents can clearly be extended to
products of three or more exponentials with the same base. The total number of factors in the
product is equal to the sum of the factors from all exponentials involved, so the exponent in the
simplified product will be just the sum of the exponents in the factors. This is illustrated in the
second example below.
examples:
57 i54 = 57+ 4 = 511
72 i75 i73 i78 = 72+5+3+8 = 718
(ii) division of one power by another:
multiply by 2 seven times
27
24
=
2x2x2x2x2x2x2
2x2x2x2
=
2x2x2
1
= 23
multiply by 2 four times
Here, the four factors of 2 in the denominator cancel four of the factors of 2 in the numerator,
leaving a net of three factors of 2 in the numerator. The denominator of 1 can simply be dropped
to get the final result 23 overall. Notice that this simplification can be written more compactly as
David W. Sabo (2003)
Laws of Exponents
Page 1 of 5
27
= 27− 4 = 23
4
2
since if we are counting up overall factors of 2 in the expression, the number of factors of 2 in the
denominator must be subtracted from the number of factors of 2 in the numerator.
In symbols, if c is any nonzero number, and m is a larger number than n, we can write
cm
= c m −n
n
c
Note that if we started with
24
27
then the denominator has more factors of 2 than does the numerator. When all possible
cancellation of factors is done, there will be three factors of 2 left on the bottom, and none on the
top:
24
1
1
= 7−4 = 3
7
2
2
2
So, in symbols, if c is any nonzero number, but now n > m, we get
cm
1
= n −m
n
c
c
examples:
58
= 58 − 6 = 5 2
56
56
1
1
= 8−6 = 2
8
5
5
5
7 2 i 7 6 7 2 + 6 78
= 3 = 3 = 78 −3 = 75
73
7
7
or
7 2 i7 6
= 7 2 + 6 −3 = 75
73
From this last example, you can see that if two or more powers with the same base are multiplied
in the numerator or the denominator or both, then the final result will have a power equal to the
sum of all exponents in the numerator minus the sum of all exponents in the denominator. This
only works for those powers that have the same base.
David W. Sabo (2003)
Laws of Exponents
Page 2 of 5
(iii) raising a power to a power:
multiply by 52 three times
(52)3 = (52) x (52) x (52)
= (5 x 5) x (5 x 5) x (5 x 5)
= 5 x 5 x 5 x 5 x 5 x 5 = 56
since 52 = 5 x 5
six factors of 5
This amounts to noting that
(5 )
2 3
= 52×3 = 56
In symbols, if c is any number, then
(c )
n m
= c n×m = c nm
In the last form in the box, we have used the algebraic convention that the product n x m can be
written simply as nm.
example:
(3 )
4 2
= 3 4×2 = 38
( whereas
3 4 i32 = 3 4+ 2 = 36 )
To summarize so far:
When a power is raised to a power you multiply the two exponents together.
When a power is multiplied by another power with the same base, you add the
exponents.
When a power is divided by another power with the same base, you subtract the second
exponent from the first.
David W. Sabo (2003)
Laws of Exponents
Page 3 of 5
(iv) raising a product to a power:
2 x 3 multiplied four times
(2 x 3)4 = (2 x 3) x (2 x 3) x (2 x 3) x (2 x 3) = 24 x 34 = 2434
four factors of 3 and four factors of 2
In general, then, if c and d are any numbers,
( cd )
n
= c nd n
example:
(5 × 3)
7
= 57 × 37
(v) raising a quotient or a fraction to a power:
If c is any number, and d is any nonzero number, then
n
cn
c
=
d 
dn
 
So, for example
5
3 3 3 3 3 3 × 3 × 3 × 3 × 3 35
3
 4  = 4 × 4 × 4 × 4 × 4 = 4 × 4 × 4 × 4 × 4 = 45
 
(vi) A Caution!
The five “laws” of exponents summarized in the square boxes above all involve multiplication or
division only. As soon as addition or subtraction occurs, the obvious relations are NOT TRUE!
For example,
But,
(2 + 3)
4
= 54 = 625
(2 + 3)
4
≠ 24 + 3 4 ,
since this latter expression evaluates to 16 + 81 = 97, which is clearly incorrect. Another example
is
David W. Sabo (2003)
Laws of Exponents
Page 4 of 5
but
(7 − 4)
5
= 35 = 243
(7 − 4)
5
≠ 75 − 45
since the latter gives 16807 – 1024 = 15783, which is clearly wrong. In both of these cases, the
rules of priority for operations (described just ahead in these topic notes) are followed in the
correct forms. It is necessary to evaluate the quantity in brackets first, and then apply the
exponent to the result. The exponents here apply to the result of doing whatever operations are
shown inside the brackets. This will become a very important rule to remember when the
brackets contain symbolic expressions rather than simple numerical expressions.
So, for powers of sums and differences, there is no simple law (such as forms (iv) and (v) above
for powers of products and quotients, respectively) that allows us to express them in terms of
powers of the original terms in the sums or differences themselves:
(c + d ) ≠ c n + d n
n
(c − d ) ≠ c n − d n
n
All of the properties of powers described in this section will become even more useful when we
deal with expressions involving symbols rather than just numbers.
David W. Sabo (2003)
Laws of Exponents
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