Exponents

Exponents are shorthand for repeated multiplication of the same thing by itself. For instance, the shorthand for multiplying three copies of the number 5 is shown on the right-hand side of the "equals" sign in (5)(5)(5) = 53. The "exponent", being 3 in this example, stands for however many times the value is being multiplied. The thing that's being multiplied, being 5 in this example, is called the "base".

This process of using exponents is called "raising to a power", where the exponent is the "power". The expression "53" is pronounced as "five, raised to the third power" or "five to the third". There are two specially-named powers: "to the second power" is generally pronounced as "squared", and "to the third power" is generally pronounced as "cubed". So "53" is commonly pronounced as "five cubed".

When we deal with numbers, we usually just simplify; we'd rather deal with "27" than with "33". But with variables, we need the exponents, because we'd rather deal with "x6" than with "xxxxxx".

Exponents have a few rules that we can use for simplifying expressions.

Simplify (x^3)(x^4)

To simplify this, I can think in terms of what those exponents mean. "To the third" means "multiplying three copies" and "to the fourth" means "multiplying four copies". Using this fact, I can "expand" the two factors, and then work backwards to the simplified form:

(x^3)(x^4) = (xxx)(xxxx)
= xxxxxxx
= x7

Note that x^7 also equals x^(3+4). This demonstrates the first basic exponent rule: Whenever you multiply two terms with the same base, you can add the exponents:

( x m ) ( x n ) = x( m + n )

However, we can NOT simplify (x^4)(y^3), because the bases are different: (x^4)(y^3) = xxxxyyy = (x^4)(y^3). Nothing combines.

Simplify (x^2)^4

Just as with the previous exercise, I can think in terms of what the exponents mean. The "to the fourth" means that I'm multiplying four copies of x2:

(x^2)^4 = (x^2)(x^2)(x^2)(x^2)
= (xx)(xx)(xx)(xx)
= xxxxxxxx
= x^8

Note that x^8 also equals x^( 2×4 ). This demonstrates the second exponent rule: Whenever you have an exponent expression that is raised to a power, you can multiply the exponent and power:

( xm ) n = x m n

If you have a product inside parentheses, and a power on the parentheses, then the power goes on each element inside. For instance, (xy2)3 = (xy2)(xy2)(xy2) = (xxx)(y^2y^2y^2) = (xxx)(yyyyyy) = x^3y^6 = (x)^3(y^2)^3. Another example would be:

For instance, given (3 + 4)^2, do NOT succumb to the temptation to say "This equals 32 + 42 = 9 + 16 = 25", because this is wrong. Actually, (3 + 4)^2 = (7)^2 = 49, not 25. When in doubt, write out the expression according to the definition of the power. Given (x – 2)2, don't try to do this in your head. Instead, write it out: "squared" means "times itself", so (x – 2)^2 = (x – 2)(x – 2) = xx – 2x – 2x + 4 = x^2 – 4x + 4.

Exponents are also called Powers or Indices

The exponent of a number says how many times to use the number in a multiplication.

In this example: 8^2 = 8 × 8 = 64
In words: 82 could be called "8 to the second power", "8 to the power 2" or simply "8 squared"



Example: 5^3 = 5 × 5 × 5 = 125
In words: 5^3 could be called "5 to the third power", "5 to the power 3" or simply "5 cubed"

Example: 2^4 = 2 × 2 × 2 × 2 = 16
In words: 2^4 could be called "2 to the fourth power" or "2 to the power 4" or simply "2 to the 4th"

Exponents make it easier to write and use many multiplications

Example: 9^6 is easier to write and read than 9 × 9 × 9 × 9 × 9 × 9
You can multiply any number by itself as many times as you want using exponents.

The exponent of a number says how many times to use the number in a multiplication.

In this example: 82 = 8 × 8 = 64
In words: 82 could be called "8 to the second power", "8 to the power 2" or simply "8 squared"


Some more examples:

Example: 53 = 5 × 5 × 5 = 125
In words: 53 could be called "5 to the third power", "5 to the power 3" or simply "5 cubed"
Example: 24 = 2 × 2 × 2 × 2 = 16
In words: 24 could be called "2 to the fourth power" or "2 to the power 4" or simply "2 to the 4th"
Exponents make it easier to write and use many multiplications

Example: 96 is easier to write and read than 9 × 9 × 9 × 9 × 9 × 9
You can multiply any number by itself as many times as you want using exponents.

In General

So, in general:

an tells you to multiply a by itself,
so there are n of those a's:


Other Way of Writing It
Sometimes people use the ^ symbol (just above the 6 on your keyboard), because it is easy to type.

Example: 2^4 is the same as 24
2^4 = 2 × 2 × 2 × 2 = 16

Negative Exponents

Negative? What could be the opposite of multiplying?

Dividing!

A negative exponent means how many times to divide one by the number.

Example: 8^-1 = 1 ÷ 8 = 0.125
You can have many divides:

Example: 5^-3 = 1 ÷ 5 ÷ 5 ÷ 5 = 0.008
But that can be done an easier way:

5^-3 could also be calculated like:
1 ÷ (5 × 5 × 5) = 1/53 = 1/125 = 0.008

In General


That last example showed an easier way to handle negative exponents:

1.Calculate the positive exponent (a^n)
2.Then take the Reciprocal (i.e. 1/a^n)

More Examples:
Negative Exponent Reciprocal of Positive Exponent Answer
4^-2 = 1 / 4^2 = 1/16 = 0.0625
10^-3 = 1 / 10^3 = 1/1,000 = 0.001
(-2)^-3 = 1 / (-2)^3 = 1/(-8) = -0.125

What if the Exponent is 1, or 0?

1 If the exponent is 1, then you just have the number itself (example 91 = 9)

0 If the exponent is 0, then you get 1 (example 90 = 1)

But what about 00 ? It could be either 1 or 0, and so people say it is"indeterminate".

Reference:
http://www.mathsisfun.com/exponent.html
http://www.purplemath.com/modules/exponent.htm