Exponents Study Guide: GED Math
Practice problems for these concepts can be found at:
RECALL THAT A FACTOR of a number is a whole number that divides evenly, without a remainder, into the given number. Frequently, you multiply the same factor times itself several times. In math, there is a special notation for this idea: exponents and the inverse operation, roots. This lesson explores powers and exponents, negative exponents, fractional exponents, scientific notation, laws of exponents, and exponents and the order of operations.
Powers and Exponents Review
Recall that when whole numbers are multiplied together, each of these numbers is a factor of the result. When all of the factors are identical, the result is called a power of that factor. For example, because 6 × 6 × 6 × 6 × 6 = 7,776, the number 7,776 is called the "fifth power of 6."
There is a shorthand notation used to indicate repeated multiplication by the same factor. This is called exponential form. In exponential form, 7,776 can be written as 65, and it is said that "six to the fifth power is 7,776." In the expression 65, the 6 is called the base and the 5 is called the exponent.
The base is the number that is used as a repeated factor in an exponential expression. It is the bottom number in an exponential expression. For example, in the expression 53, 5 is the base number.
The exponent indicates the number of times that a repeated factor is multiplied together to form a power; it is the superscript in an exponential expression. In the expression 53, the 3 is the exponent and indicates that 5 is multiplied by itself twice.
The power is a product that is formed from repeated factors multiplied together. For example, 27 is the third power of 3, because 3 × 3 × 3 = 27.
Observe the following pattern for the base of 3:
Look at the last column, standard form. As you go down this column, each time the entry is multiplied by 3 to obtain the next entry below, such as 81 × 3 = 243. Notice that as you go up this column, each time you would divide the entry by 3 to get the entry above. Now extend this pattern to include an exponent of zero and some negative exponents:
From the patterns noted, the rules for zero exponents and negative exponents follow:
- Any number (except zero) to the zero power is 1. x0 = 1, where x is any number not equal to zero. For example, 70 = 1, 290 = 1, and 2,1590 = 1.
- A negative exponent is equivalent to the reciprocal of the base, raised to that positive exponent. , where x is not equal to zero. Recall that the reciprocal of 4 is . So, for example, , and , and .
When an exponent is a fraction, the denominator of this fractional exponent means the root of the base number, and the numerator means a raise of the base to that power.
numerator—the power to which the base is raised
denominator—the root of the base number
For example, 81/3 is the same as = 2. Another example is 82/3 means , which is = 4, or alternatively, , which is 22 = 4.
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