Lesson EXPONENTS

Source code of 'EXPONENTS'

This Lesson (EXPONENTS) was created by by Theo(13342)

This lesson covers an overview of EXPONENTS.

REFERENCES

<a href = "http://www.purplemath.com/modules/exponent.htm" target = "_blank">http://www.purplemath.com/modules/exponent.htm</a>
<a href = "http://www.algebrahelp.com/lessons/simplifying/numberexp/" target = "_blank">http://www.algebrahelp.com/lessons/simplifying/numberexp/</a>
<a href = "http://tutorial.math.lamar.edu/Classes/Alg/IntegerExponents.aspx" target = "_blank">http://tutorial.math.lamar.edu/Classes/Alg/IntegerExponents.aspx</a>
<a href = "http://tutorial.math.lamar.edu/Classes/Alg/RationalExponents.aspx" target = "_blank">http://tutorial.math.lamar.edu/Classes/Alg/RationalExponents.aspx</a>
<a href = "http://tutorial.math.lamar.edu/Classes/Alg/RealExponents.aspx" target = "_blank">http://tutorial.math.lamar.edu/Classes/Alg/RealExponents.aspx</a>
<a href = "http://www.wtamu.edu/academic/anns/mps/math/mathlab/col_algebra/col_alg_tut5_ratexp.htm" target = "_blank">http://www.wtamu.edu/academic/anns/mps/math/mathlab/col_algebra/col_alg_tut5_ratexp.htm</a>
<a href = "http://www.wtamu.edu/academic/anns/mps/math/mathlab/col_algebra/col_alg_tut2_exp.htm" target = "_blank">http://www.wtamu.edu/academic/anns/mps/math/mathlab/col_algebra/col_alg_tut2_exp.htm</a>
<a href = "http://www.mathsisfun.com/algebra/exponent-laws.html" target = "_blank">http://www.mathsisfun.com/algebra/exponent-laws.html</a>
<a href = "http://oakroadsystems.com/math/expolaws.htm" target = "_blank">http://oakroadsystems.com/math/expolaws.htm</a>

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:
{{{5^3 = 125}}}

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

{{{x^n}}}

This means you take x and multiply it by itself (n-1) times.

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

Example:
{{{2^5}}} = 2 multiplied by itself 4 times = {{{2*2*2*2*2}}} = 32

DEFINITION NUMBER 2

{{{root(n,x)}}}

This means you find a number that, when raised to the power of n, equals x.

Example:
{{{root(5,32)}}} = 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

{{{x^n = y}}} if and only if {{{root(n,y) = x}}}

This means that if you raise x by the power of n to get y, then you have to be able to lower y by the root of n to get back to x.

The power of the root must equal the power of the exponent.

Example:
{{{2^5 = 32}}} if and only if {{{root(5,32) = 2}}}

This means that 2 raised to the power of 5 = 32 if and only if 32 lowered by the root of 5 = 2.

DEFINITION NUMBER 4

{{{x^0}}} = 1

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

Example:
{{{99999999^0 = 1}}}

DEFINITION NUMBER 6

{{{x^(1/n) = root(n,x)}}}

This means that raising something to the power of 1/n is the same thing as reducing that something by the root of n.

Example:
{{{125^(1/3) = root(3,125)}}} = 5

RULES OF EXPONENT OPERATIONS

RULE NUMBER 1

{{{(x^m)*(x^n)}}} = {{{x^(m+n)}}}

Example:
x = 5
m = 2
n = 3

{{{x^m*x^n}}} = {{{x^(m+n)}}} becomes:
{{{(5^2)*(5^3)}}} = {{{5^(3+2)}}} = {{{5^5}}} = {{{3125}}}

since
{{{5^2 = 25}}} and {{{5^3 = 125}}} and {{{25*125}}} = {{{3125}}} 
the rule is confirmed to be true.

RULE NUMBER 2

{{{(x^m)/(x^n)}}} = {{{x^(m-n)}}}

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

{{{(x^m)/(x^n)}}} = {{{x^(m-n)}}} becomes:
{{{(7^5)/(7^2)}}} = {{{7^(5-2)}}} = {{{7^3}}} = {{{343}}}

since
{{{7^5 = 16807}}} and {{{7^2 = 49}}} and {{{16807/49}}} = {{{343}}}
the rule is confirmed to be true.

RULE NUMBER 3

{{{(x^m)^n}}} = {{{x^(m*n)}}}

Example:
x = 15
m = 3
n = 2

{{{(x^m)^n}}} = {{{x^(m*n)}}} becomes:
{{{((15)^3)^2}}} = {{{(15)^(2*3)}}} = {{{(15)^6}}} = {{{11390625}}}

since
{{{(15)^3 = 3375}}} and {{{(3375)^2}}} = {{{11390625}}}
the rule is confirmed to be true.

RULE NUMBER 4

{{{(x*y)^n}}} = {{{(x^n)*(y^n)}}}

Example:
x = 26
y = 3
n = 2

{{{(x*y)^n}}} = {{{(x^n)*(y^n)}}} becomes:
{{{(26*3)^2}}} = {{{(26^2)*(3^2)}}} = {{{(676)*(9)}}} = {{{6084}}}

since
{{{26*3 = 78}}} and {{{78^2}}} = {{{6084}}}
the rule is confirmed to be true.

RULE NUMBER 5

{{{((x^m)*(y^n))^p}}} = {{{((x^m)^p) * ((y^n)^p)}}} = {{{(x^(m*p)) * (y^(n*p))}}}

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

{{{((x^m)*(y^n))^p}}} = {{{((x^m)^p) * ((y^n)^p)}}} = {{{(x^(m*p)) * (y^(n*p))}}} becomes:
{{{((2^3)*(3^2))^4}}} = {{{((2^3)^4) * ((3^2)^4)}}} = {{{(2^(3*4)) * (3^(2*4))}}} = {{{(2^12) * (3^8)}}} = {{{4096 * 6561}}} = {{{26873856}}}

since
{{{2^3 = 8}}} and {{{3^2 = 9}}} and {{{8*9 = 72}}} and {{{72^4}}} = {{{26873856}}}
the rule is confirmed to be true.

RULE NUMBER 6

{{{(x/y)^n}}} = {{{(x^n)/(y^n)}}}

Example:
x = 4
y = 2
n = 5

{{{(x/y)^n}}} = {{{(x^n)/(y^n)}}} becomes:
{{{(4/2)^5}}} = {{{(4^5)/(2^5)}}} = {{{1024/32}}} = {{{32}}}

since
{{{(4/2)^5}}} = {{{2^5}}} = {{{32}}}
the rule is confirmed to be true.

RULE NUMBER 7

{{{x^(m/n)}}} = {{{(root(n,x))^m}}} = {{{root(n,x^m)}}}

the exponent on the left side of the equation is (m/n).

Example:
x = 4
m = 5
n = 2

{{{x^(m/n)}}} = {{{(root(n,x))^m}}} = {{{root(n,x^m)}}} becomes:
the exponent on the left side of the equation is (m/n).

{{{4^(5/2)}}} = {{{(root(2,4))^5}}} = {{{root(2,4^5)}}}
{{{4^(5/2)}}} = {{{4^(2.5)}}} = 32
the exponent on the left side of the equation is (5/2).

{{{(root(2,4))^5}}} = {{{2^5}}} = 32
{{{root(2,4^5)}}} = {{{root(2,1024)}}} = 32
since all methods get the same answer, the rule is confirmed.

RULE NUMBER 8

{{{x^(-n))}}} = {{{1/(x^n)}}}

This means that x raised to the power of -n is equal to 1 divided by (x raised to the power of n).

Example:
{{{2^(-5))}}} = {{{1/(2^5)}}} = {{{1/32}}}

SIMPLIFICATION

The goal of simplification when it comes to exponents is to:

be left with:
1.  only positive exponents.
2.  the same base occurring only once in the expression.
3.  all constants being reduced to their simplest terms

Examples:
1.  {{{x^2 * x^3}}} = {{{x^(2+3)}}} = {{{x^5}}}

2.  {{{x^3 / x^2}}} = {{{x^(3-2)}}} = {{{x}}}

3.  {{{x^(-5)}}} = {{{1/(x^5)}}}

4.  {{{(x^2)^3}}} = {{{x^(2*3)}}} = {{{x^6}}}

5.  {{{(x/y)^5 / (x/y)^2)}}} = {{{(x/y)^(5-2)}}} = {{{(x/y)^3}}} = {{{x^3/y^3}}}

6.  {{{((x*y)^5)/((x*y)^3)}}} = {{{(x*y)^(5-3)}}} = {{{(x*y)^(2)}}} = {{{(x^2)*(y^2)}}}

7.  {{{x^(1/5) * x^(4/5)}}} = {{{x^((1/5) + (4/5))}}} = {{{x^(5/5)}}} = {{{x^1}}} = {{{x}}}
first exponent of x is (1/5).
second exponent of x is (4/5).
exponents are added together to get (1/5) + (4/5) = (5/5) = 1.

REMOVING NEGATIVE EXPONENTS FROM RATIONAL EXPRESSIONS

a rational expression raised to a negative power would be something like:

{{{(x/y)^(-n)}}}
to make the exponent positive, this becomes equivalent to:
{{{1/(x/y)^n}}}

this is equivalent to:
{{{1/(x^n/y^n)}}}
if we let a = x^n and we let b = y^n, then we get:
{{{1/(a/b)}}}

if we multiply both numerator and denominator of this equation by b, then we get:
{{{1/(a/b)}}} = {{{(1*b)/((a/b)*b)}}} which becomes:

{{{b/a}}}

if we now replace a with x^n and b with y^n, then we get:
{{{(y^n)/(x^n)}}}
this is equivalent to:
{{{(y/x)^n}}}

our original equation of:
{{{(x/y)^(-n)}}}
has become:
{{{(y/x)^n}}}

an example of how this works in practice would be:

{{{(6/2)^(-2)}}} = {{{(2/6)^2}}} which becomes:
{{{(2^2/6^2)}}} which becomes:
{{{4/36}}} which becomes:
{{{1/9}}}

we confirm by using our calculator to calculate the result directly.
{{{(6/2)^(-2)}}} is placed in the calculator as (6/2)^(-2) to get an answer of .11111111.
we multiply that answer by 9 to get 1
.1111111111 is equivalent to 1/9
this is the same answer that we got by applying the technique shown above.

This leads to rule number 9 as shown below:

RULE NUMBER 9

{{{(x/y)^(-n)}}} = {{{(y/x)^n}}}

example:

x = 2
y = 3
n = 2

{{{(x/y)^(-n)}}} = {{{(y/x)^n}}} becomes:
{{{(2/3)^(-2)}}} = {{{(3/2)^2}}} which becomes:
{{{3^2/2^2}}} which becomes:
9/4

checking with your calculator should confirm this answer is correct.
enter into your calculator:
(2/3)^(-2) to get an answer of 2.25
to convert to a fraction with a denominator of 4, multiply the answer by 4 to get a numerator of 9 and a denominator of 4 which equals 9/4.

SPECIAL ON CONVERTING A DECIMAL TO A FRACTION USING YOUR CALCULATOR.

multiply the decimal equivalent by the denominator of the fraction you are looking for.
the result should be the numerator of the fraction you are looking for.

example:

you started with 3/4
the decimal equivalent is .75
you want to find the equivalent fraction with a denominator of 4.
you multiply the decimal equivalent by 4 and you get a numerator of 3.
3/4 converted to decimal is determined by dividing 3 by 4.
(3/4) in decimal format is converted to fraction format with the original denominator by multiplying the decimal equivalent by 4 to get the numerator of the fraction.
(3/4) * 4 = 3
the decimal equivalent of (3/4) is equal to .75
.75 * 4 = 3

if you wind up with a numerator that's close to an integer but not quite right on, then confirm that the integer is indeed the numerator of the fraction by doing the following:
7/3 = 2.3
2.3 * 3 = 6.9
it looks like the numerator should be 7 but we're not quite sure.
assume it is 7 and find the decimal equivalent of 7/3.
you get 2.333333333333333
since the decimal equivalent you were looking at is 2.3, you can be reasonably confident that 2.3 is the rounded version of 2.33333333333333 and that the numerator of the equivalent fraction with a denominator of 3 is 7.