Algebra: Exponents

This section will help you understand the notation used to indicate powers, or exponents.

Important Things to Remember

• Exponents are a "short cut" method of showing a number is multiplied by itself.
• Exponents are indicated as follows:

                 x^2

  They can also be indicated by superscripted characters.
• Know the difference between -x^y and (-x)^y.

                 -3^4 = -(3)(3)(3)(3) = -81
                 (-3)^4 = (-3)(-3)(-3)(-3) = 81

The Tutorial
When dealing with exponents, remember that exponents are a "short cut" to show that a number is to be multiplied by itself a given umber of times.  For example, x^2 is the same as x * x.  The number or symbol (variable) that is to be multiplied by itself is called the base (in the example given above, the base is x), and the number or symbol showing how many times it is to be multiplied by itself is called the exponent or power (in the example above, the power is 2).

Examples:

1. Simplify: (-5)^3
   Solution: (-5)(-5)(-5)        Take note of the parentheses.  Realize that the
             -125                the problem is (-5) cubed.  The power (3) shows
                                 that the base [ (-5) ] needs to be multiplied by
                                 itself 3 times.

2. Simplify: -3^3 - (-3)^2 + (-2)^2
   Solution: -(3)(3)(3) - (-3)(-3) + (-2)(-2)       Watch out for the first term,
             -27 - 9 + 4                            which does not have
             -32                                    parentheses around it.
                                                    Simplify each expression and
                                                    then add the terms for the
                                                    final answer.
         

This section will help you understand how to evaluate expressions with exponents.

Important Things to Remember

• Be sure to note parentheses when dealing with exponents.  Always evaluate anything inside parentheses first.  Example:

                 -3^4 = -(3)(3)(3)(3) = -81
                 (-3)^4 = (-3)(-3)(-3)(-3) = 81

The Tutorial
Evaluating expressions is something very common in algebra and is useful later on when you have to check solutions of equations.  Evaluating expressions with exponents is just as easy.  Just be sure to notice negative numbers and negative signs when dealing with exponents because they can make a big difference.

Example:

1. Evaluate: yx^2z^3
             y = 3, x = 4, z = 2
   Solution: (3)(4)^2(2)^3       Plug the numbers into the expression and
             (3)(16)(8)          simplify.
             384
         

This section will help you better understand these theorems.

Important Things to Remember

• x^0 = 1 when x does not equal 0.  x can be anything (except 0), including numbers, variables, or an equation.
• x^1 = x
• x^(-n) = 1/(x^n) when x does not equal 0.  Example:

                 x^(-2) = 1 / (x^2)

• Answers are considered simplified when exponents are positive unless otherwise noted.

The Tutorial
There are three theorems that are special to exponents.  They are outlined below.

Product Theorem for Exponents
If m and n are real numbers and x does not equal 0, x^m * x^n = x^(m + n)

Quotient Theorem for Exponents
If m and n are real numbers and x does not equal 0, (x^m) / (x^n) = x^(m - n) = 1/(x^(n - m))

Power Theorem for Exponents
If m and n are real numbers and x does not equal 0, (x^m)^n = x^(mn)

Examples:

1. Simplify: x^2y^2x^5y^3
   Solution: x^2x^5y^2y^3        Rearrange the factors so they are easier to deal
             x^7y^5              with.  Use the Product Theorem to simplify the
                                 expression.

2. Simplify: x^4
             ---
             x^6
   Solution:    1                Use the Quotient Theorem to combine the
             -------             numerator and denominator into one term in the
             x^(6-4)             denominator.

              1
             ---
             x^2

3. Simplify: (x^(-4))^(-2)        
   Solution: x^8                 Use the Power Theorem to multiply the two
                                 exponents into one.
         

This section will help you understand how to combine like terms when they include exponents.

Important Things to Remember

• Rearranging the order of the factors does not change their value.  Example:

                 x^2yz^5 = z^5x^2y

• Power Theorem for Exponents, which states (x^m)^n = x^(mn)

The Tutorial
When you come across an expression with many terms, it is easier to deal with that expression when it has been simplified by adding like terms.  When doing this with an expression that contains exponents, the variables and their exponents have to be the same.

Example:

1. Simplify: x^2yz^5 + 2xy^2z^5 + 3z^5x^2y - 7y^2xz^5
   Solution: x^2yz^5 + 3x^2yz^5 + 2xy^2z^5 - 7xy^2z^5         Rearrange the
             4x^2yz^5 - 5xy^2z^5                              factors so they
                                                              are more easily
                                                              identifiable as
                                                              like terms.
                                                              Combine like
                                                              terms and get
                                                              the answer.
         

Take the quiz on exponents.  The quiz is very useful for either review or to see if you've really got the topic down.