Property of Exponents

Properties of Exponents

In this module we will cover the Properties of Exponents.

Exponents are a large part of mathematics and you need to memorize each of the rules.
Rules for Working with Exponents
Let "a" be a real number:
an = a x a x a...... n times
a0 = 1
a-n = 1/a+n
Product of Powers:
(a)m・(a)n = am + n
Power of a Power:
(am)n = am・n
Power of a Product:
(ab)m = am・bm
Quotient of Powers:
am/ an = am - n
Power of a Quotient:
(a/b)m = am / bm
Let's look at a visual way to interpret exponent rules:
Simplify: x3y2(x3)2
can be modeled using shapes:
ex
Example 2: Simplify
x2y4(y2)3
ex2
Example 3: Simplify
x4(y3)2(z2)3 ex3
Example 4: Simplify
x2y3(x2)4z4 ex4
Example 5: Simplify
x3y4(y2)3z2
Solution:
Group like terms: x3y4y2y2y2z2
Combine like terms: x3y10z2 *Answer
Example 6: Simplify
x0(yz)3
ex5
Example 7: Simplify
(x2/y3)3
ex 7
Now you try a few examples:
(x2)4(y3)2z4
Use the caret key (^) to type in your exponents.
Do not leave empty spaces between the terms.


Simplify:
(x2y4)5(z3)2


Simplify:
(x3/y2)5(z3)4


Examples of the rule a-n = 1 / an
When you simplify exponent equations, you cannot have negative exponents with your answers. So let's look at the concept of how to eliminate negative exponents:
We will use the "Product of Powers" rule.
Example: x-2・ x2 = x-2 + 2 = x0 = 1
So multiplying the exponent by its opposite power eliminates the term. When dealing with negative exponents they will normally be part of a fraction. So when we multiply the numerator by a value, we must also multiply the denominator by the same value to keep the equation balanced.
Example: Simplify: x-2/y-3
ex8
Example 2: Simplify: x-5/y-4
ex9
Example 3: Simplify: x-6/y-3
ex10
Notice the pattern between the original problems and their simplified answers:
ex11 After all the steps to simplify, can you think of a way to do the problem just by following the pattern?

Examples:
ex12 When simplifying exponential expressions, you must not have any negative exponents in your answer.
Examples:
ex13