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This page is designed to help you better understand how to deal with exponents and their uses in algebra.  (Once you learn something in algebra, you've learned it for any higher math - and you will use it!)  Click any of the links below or scroll down to start understanding exponents! Exponential notation Evaluating exponentials Theorems for exponents (product, quotient, and power) Like terms with exponents Quiz on Exponents Exponents, or powers, are an important part of math as they are necessary to indicate that a number is multiplied by itself for a given number of times.  This section will help you understand the notation used to indicate powers, or exponents.  Exponents are a "short cut" method of showing a number is multiplied by itself.  Exponents can be shown to different ways.  Example:          x2 or x^2 Know the difference between -x^y and (-x)^y.  Example:          -34 = -(3)(3)(3)(3) = -81         (-3)4 = (-3)(-3)(-3)(-3) = 81 When dealing with exponents, remember that exponents are a "short cut" to show that a number is to be multiplied by itself a given number of times.  For example, x2 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).  1.   Simplify:   (-5)^3      Solution:        (-5)(-5)(-5)         -125   Take note of the parentheses. Realize that the problem is (-5) cubed. The power (3) shows that the base (-5) needs to multiplied by itself 3 times. 2.   Simplify: -33 - (-3)2 + (-2)2      Solution:        -(3)(3)(3) - (-3)(-3) + (-2)(-2)      -27 - 9 + 4      -32   Watch out for the first term, which does not have parentheses around it. Simplify each expression and then add the terms for the final answer. Back to Top  The evaluation of expressions containing exponents is very straightforward.  It is the same as the evaluation of any other expression.  The only thing to look out for is a negative number. This section will help you understand how to evaluate expressions with exponents.  Be sure to note parentheses when dealing with exponents.  Always evaluate anything inside parentheses first.  Example:          -34 = -(3)(3)(3)(3) = -81         (-3)4 = (-3)(-3)(-3)(-3) = 81 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. 1.   Evaluate:   yx2z3 y = 3, x = 4, z = 2      Solution:        (3)(4)2(2)3      (3)(16)(8)      384   Plug the numbers into the expression. Simplify. Back to top  There are three different theorems that deal with exponents. This section will help you better understand these theorems. x0 = 1 when x does not equal 0.  x can be anything (except zero), including numbers, variables, or an equation. x1 = x x-n = 1/xn when x does not equal 0.  Example:        x-2 = 1 / x2 Answers are considered simplified when exponents are positive unless otherwise noted.  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, xm * xn = xm+n Quotient Theorem for Exponents If m and n are real numbers and x does not equal 0, xm/xn = xm-n = 1/xn-m Power Theorem for Exponents If m and n are real numbers and x does not equal 0, (xm)n = xmn 1.   Simplify:   x2y2x5y3      Solution:        x2x5y2y3           x7y5   Rearrange the factors so they are easier to deal with. Use the Product Theorem to simplify the expression. 2.   Simplify:   x4 -- x6      Solution:        1      ----      x6-4           1      ---      x^2   Use the Quotient Theorem to combine the numerator and denominator into one term in the denominator. 3.   Simplify:   (x-4)-2      Solution:        x8   Use the Power Theorem to multiply the two exponents into one. Back to top  Exponents add a new aspect to the operation of combining like terms, and thus make it slightly more complicated. This section will help you understand how to combine like terms when they include exponents. Rearranging the order of factors does not change their value.  Example:         x2yz5 = z5x2y Power Theorem for Exponents, which states (xm)n = xmn. 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. 1.   Simplify:   x2yz5 + 2xy2z5 + 3z5x2y - 7y2xz5      Solution:        x2yz5 + 3x2yz5 + 2xy2z5 - 7xy2z5           4x2yz5 - 5xy2z5   Rearrange the factors so they are more easily identifiable as like terms. Combine like terms and get this answer. Back to top  Take the Quiz on exponents.  (Very useful to review or to see if you've really got this topic down.)  Do it!