Exponents
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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