Powers and Radicals
Exponents
When a number is raised to a power, it indicates how many times it is multiplied with itself. The exponent is used to represent the number of times a number or variable is multiplied with itself. The exponent is either a superscript such as x2 or a carat or hat symbol ^ that is used to indicate that the number or variable following it is an exponent, x^2. Exponents provide a way to make an expression shorter and more compact. It is more convenient to write x5 or x^5 than xxxxx.
In algebra, it is important to be able to work with exponents. Like many other concepts in math, there is a set of rules that tell how to calculate expressions that contain exponents.
| a0 = 1 | am * an = am+n | |
| a-m = 1/am | (am)n = amn | |
| am = 1/a-m | am/an = am-n if m > n | |
| a1/n = n√a | am/an = 1 if m = n | |
| am/n = n√am = (n√a)m | am/an = 1/an-m if n > m | |
| (a/b)-n = (b/a)n | (ab)m = ambm | |
| alogab = b | (a/b)m = am/bm |
Radicals
The radical can be thought of as the inverse of the exponent. While the exponent represents the number of times something is multiplied with itself, the radical represents the number of times a factor is multiplied to obtain the number under the radical. When an expression is under a radical and has an exponent, the radical in some cases can be thought of as undoing the exponent. Example: √x2 = x. The radical itself can be seen as a fractional exponent: √x = x(1/2).
Radical Rules and Properties
| b(m/n) = (b(1/n))m | n√(a*b) = n√an√b | |
| b(m/n) = (b(1/n))m | (a*b)(1/n) = a(1/n)b(1/n) | |
| b(m/n) = (bm)(1/n) | n√(a/b) = (n√a) / (n√b) | |
| b(m/n) = n√bm | mn√an = m√a | |
| b(m/n) = (n√b)m | n√m√a = mn√a | |
| √b2 = |b| b is a real number | n√am√b = mn√(ambn) |
Conversion
In algebra as well as other math, it is important to convert an expression from one form to another.
