Lesson EXPONENTS

This lesson covers an overview of EXPONENTS.

REFERENCES

http://www.wtamu.edu/academic/anns/mps/math/mathlab/col_algebra/index.htm
http://www.purplemath.com/

These references have lots more example and practice problems as well. There are other excellent references online as well. Paul’s Online Notes and Dr. Math are ones that I have used in the past, among others. To find additional help online, go to www.yahoo.com or www.google.com and search for the subject of interest.

TERMINOLOGY

An exponent is what is used to raise something to a higher power. You would normally reference it as something cubed or something squared or something to the 15th power.

Example:

This means 5 cubed = 125./
5 is the base used in the operation.
3 is the exponent used in the operation.
125 is the result of the operation.

DEFINITIONS

Exponents and roots are intimately tied together. It is difficult to talk fully about one without talking about the other. That is why roots are being discussed here as well. For more information on roots, see the lesson on RADICALS

DEFINITION NUMBER 1

This means you take x and multiply it by itself (n-1) times.
n is assumed to be an integer > 0

x is the base. It is what is being raised by the exponent.
n is the exponent.

Example:

= 2 multiplied by itself 4 times =

= 32

DEFINITION NUMBER 2

This means you find a number that, when raised to the power of n, equals x.
n is assumed to be an integer > 0

Example:

= 2

The radicand is 32. The index of the radicand or the power of the root is 5. The result of the operation or the root is 2. You would say that the 5th root of 32 = 2.

DEFINITION NUMBER 3

if and only if

This means that if you can raise something to a certain power, then you have to be able to lower the result by a root to the original something. The power of the root must equal the power of the exponent.

Example:

if and only if

This means that 2 to the fifth power = 32 if and only if the 5th root of 32 = 2.

DEFINITION NUMBER 4

= 1

This means that any number to the zero power is equal to 1.

Example:

DEFINITION NUMBER 5

This means that any number raised to a negative power is equal to it's reciprocal raised to a positive power.

Example:

DEFINITION NUMBER 7

This means that raising something to the (1/n)th power is the same thing as reducing that something to nth root.

Example:

= 5

This means that raising 125 to the one third power equals taking the third root of 125 equals 5.

RULES

RULE NUMBER 1

Example:
x = 5
m = 2
n = 3

becomes:

since

and

the rule is confirmed to be true.

RULE NUMBER 2

An example of this would be:
x = 7
m = 5
n = 2

becomes:

since

and

the rule is confirmed to be true.

RULE NUMBER 3

Example:
x = 15
m = 3
n = 2

becomes:

since

and

the rule is confirmed to be true.

RULE NUMBER 4

Example:
x = 26
y = 3
n = 2

becomes:

since

and

the rule is confirmed to be true.

RULE NUMBER 5

Example:
x = 2
m = 3
y = 3
n = 2
p = 4

becomes:

since

and

the rule is confirmed to be true.

RULE NUMBER 6

Example:
x = 4
y = 2
n = 5

becomes:

since

the rule is confirmed to be true.

RULE NUMBER 7

Example:
x = 4
m = 5
n = 2

becomes:

= 32

= 32
since all methods get the same answer, the rule is confirmed.

SIMPLIFICATION

The goal of simplification when it comes to exponents is to:
1. Be left with only positive exponents.
2. Be left with the same base occurring only once in the expression.
3. Be left with all constants being reduced to their simplest terms

Examples:
1.