Exponents
A Code for Multiplication
A Code for Multiplication
The Secret to understanding and working with exponents is to think of them as a Code
for Multiplication. For example: This code for 2*2*2 = 8 is 23 = 8.
Therefore, 23 when it is decoded means to 'multiply 2 by itself 3 times'. Using this code the number to be multiplied by itself (2) is called
the base and the number of times this base number is multiplied
by itself (3) is called the exponent or alternatively the 'power' of 3.
Likewise, the Code for 3*3*3*3 = 81 is 34 = 81 with a base of 3 raised to the power of 4. Two other examples with numbers are given. 53 has a base 5 raised to the power of 3 and decoded
means 5*5*5 = 125. 103 has a base 10 raised to the power of 3 and
decoded means 10*10*10 = 1,000.
The base can be a variable like x as well as a number. Two examples of
this are given in the picture above. x*x*x encoded in our Multiplication Code
is x3 which has a base of x raised to the power of 3. Likewise,
x*x*x*x is encoded as x4 with a base of x
raised to the power of 4.
So, we see that Exponents are just a fast, coded way to do
multiplication.
Multiplication Code - Negative Powers
Our Multiplication Code is also valid for negative exponents with one
additional encryption rule. A negative power is encoded the same as
a positive power except that multiplication is done in the
denominator of a fraction with 1 in the numerator.
For example 1/(2*2*2) is encoded as 1/(23) = 1/8 which is encoded as 2-3 = 1/8. Likewise,
3-4 when decoded means 1/(34) = 3-4 and
so on for 5-3 and 10-3 as shown in the picture
above.
As for positive powers, negative powers may be used for variables
like x as well as numbers. Thus, x-3 when decoded =
1/(x3 = 1/(x*x*x) and x-4 when decoded =
1/(x4) = 1/(x*x*x*x) as shown in the figure above.
Product Rule for Powers
There are seven rules that define how Exponents operate and are can
be used as coded shortcuts to solve many problems involving exponents.
The first of these 7 rules is called the Product Rule and is shown in
the picture above.
The Product Rule applies when we multiply numbers or variables with
exponents. Using our Multiplication Code we can express
23*24 as (2*2*2)*(2*2*2*2) which is =
(2*2*2*2*2*2*2). When we apply the Code to this product we see
that it = 27. Since 7 can be expressed as (3+4) we see that
23*24 = 2(3+4).
As seen in the picture above the same steps can be taken to show that
x3*x4 = x(3+4).
The generalized Product Rule for Powers is
xm*xn = x(m+n)
Power Rule for Exponents
The second of the seven Rules of Exponents is the Power Rule shown in the picture above.
The Power Rule applies when we find numbers or variables with
exponents which are themselves also raised to a power or exponent.
Using our Multiplication Code we can express (23)4 as
(2*2*2)*(2*2*2) * (2*2*2)*(2*2*2) which is = (2*2*2*2*2*2*2*2*2*2*2*2).
When we apply the Code to this product we see that it =
212. Since 12 can be expressed as (3*4) we see that
(23)4 = 2(3*4).
As seen in the picture above the same steps can be taken to show that
(x3)4 = x(3*4).
The generalized Power Rule for Exponents is
(xm)n = x(m*n)
Product Rule with 2 Bases
The third of the seven Rules is the Product Rule for 2
Bases as shown in the picture above.
The Product Rule for 2 bases is the same as the Product Rule except one
of the numbers or variables involved in the product has a different
base. Using our Multiplication Code we can express
23*34 as (2*2*2)*(3*3*3*3) which cannot be combined is simply =
23*34.
As seen in the picture above the same steps can be taken to show that
x3*y4 = x3*y4.
The generalized Product Rule with 2 Bases is
xm*yn = xm*yn
Power Rule with 2 Bases
The fourth of the seven Rules is Power Rule for 2 Bases
shown in the picture above.
The Power Rule for two bases is the same as the Power Rule except one of
the numbers or variables involved in the product raised to a power has a
different base.
Using our Multiplication Code we can express
(22*33)4 as (2*2*2*2*2*2*2*2) *
(3*3*3*3*3*3*3*3*3*3*3*3) which is = (28)*312.
Since 8 can be expressed as (2*4) and 12 can be expressed as (3*4) we
see that (22*33)4 =
28*312 which is = 2(2*4)*3(3*4).
As seen in the picture above steps can be taken to show that
(x2*y3)4 = x8*y12 which is = x(2*4)*y(3*4).
The generalized Power Rule is
(xm*yn)z = x(m*z)*y(m*z)
Negative Power Rule
As discussed above our Multiplication Code is also valid for negative powers with one
additional encryption rule. A negative power is encoded the same as
a positive power except that multiplication is done in the
denominator of a fraction with 1 in the numerator.
As shown in the picture above 1/(2*2*2) is encoded as 1/(23) = 1/8 which in
our code is encoded as 2-3 = 1/8.
Negative powers may also be used for variables
like x as well as numbers. Thus, x-3 when decoded =
1/(x3) = 1/(x*x*x) as shown in the figure above.
The generalized Negative Rule is
x-m = 1/(xm)
The Quotient Rule
The Quotient Rule applies when we divide numbers or variables raised to various powers. Using our Multiplication Code we can express
24/23 as (2*2*2*2)/(2*2*2) which is =
(16)/(8) = 2 = 21 = 2(4-3).
As seen in the picture above the same steps can be taken to show that
x4/x3 = x(4-3).
The generalized Product Rule is
xm/xn = x(m-n)
The Distributive Rule for Powers
The Distributive Rule specifies which arithmetic operations exponents distribute over. The general statement of this rule is
Exponents DO Distribute across multiplication and division
Exponents DO NOT Distribute across addition and subtraction.
This can be seen in the following number examples.
For Multiplication:
(23*32)2 = (2(3*2)*3(2*2))
= (26*34) = (64*81) = 5,184
(23*32)2 = (8*9)2 = (72)2 = 5,184
and for Addition:
(23 + 32)2 = (2(3*2) + 3(2*2))
= (26 + 34) = 64 + 81 = 145
(23 + 32)2 = (8 + 92 = (17)2
= 289 NOT = 145
Summary - The 7 Rules
The 7 Rules are summarized in the picture above
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