Practice problems for these concepts can be found at:
Exponents and Roots Practice Problems: GED Math
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.
Negative Exponents
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 
.
Fractional Exponents
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.
Scientific Notation
Scientists measure very large numbers, such as the distance from the earth to the sun, or very small numbers, such as the diameter of an electron. Because these numbers involve a lot of digits as placeholders, a special notation was invented as a shorthand for these numbers. This scientific notation is very specific in the way it is expressed:
where n = a number greater than 1 and less than 10
To find the scientific notation of a number greater than 10, locate the decimal point and move it either right or left so that there is only one non-zero digit to its left. The result will produce the n part of the standard scientific notational expression. Count the number of places that you had to move the decimal point. If it is moved to the left, as it will be for numbers greater than 10, that number of positions will equal the e part.
Example
What is 23,419 in scientific notation?
Position the decimal point so that there is only one non-zero digit to its left: 2.3419
Count the number of positions you had to move the decimal point to the left, and that will be e: 4.
In scientific notation, 23,419 is written as 2.3419 × 104.
For numbers less than 1, follow the same steps except in order to position the decimal point with only one non-zero decimal to its left, you will have to move it to the right. The number of positions that you move it to the right will be equal to –e. In other words, there will be a negative exponent.
Example
What is 0.000436 in scientific notation?
First, move the decimal point to the right in order to satisfy the condition of having one non-zero digit to the left of the decimal point: 4.36
Then, count the number of positions that you had to move it: 4.
That will equal –e, so –e = –4, and 0.000436 = 4.36 × 10–4.
For numbers already between 1 and 10, you do not need to move the decimal point, so the exponent will be zero.
Laws of Exponents
When working with exponents, there are three rules that can be helpful.
- When you multiply powers with the same base, keep the base and add the exponents: 42 × 43 = 42 + 3 = 45.
- When you divide powers with the same base, keep the base and subtract the exponents: 37 ÷ 34 = 37 – 4 = 33.
- When you raise a power to a power, you keep the base and multiply the exponents: (73)2 = 73 × 2 = 76.
If two powers are being multiplied together and the bases are not the same, check to see if you can convert the numbers to have the same base to use the preceding laws. For example, to simplify 272 × 32, recognize that 27 can be written as 33. Change the problem to (33)2 × 32. Use law number 3 to get 33 × 2 × 32 = 36 × 32. Now, you can use law number 1 to get 36 + 2 = 38.
Exponents and the Order of Operations
Be aware of some distinctions when working with the order of operations and exponents. Exponents are done after parentheses and before any other operations, including the negative sign. For example, –32 = –(3 × 3) = –9 because you first take the second power of 3 and then the answer is negative. However, (–3)2 = –3 × –3 = 9, because –3 is enclosed in parentheses.
Example
–20 + (–2 + 5)3 ÷ (10 – 7) × 2 =
Evaluate the parentheses, left to right:
–20 + 33 ÷ 3 × 2.
Now, evaluate the exponent: –20 + 27 ÷ 3 × 2.
Division will be done next: –20 + 9 × 2.
Evaluate multiplication: –20 + 18.
Finally, perform addition to arrive at the answer: –2.
Practice problems for these concepts can be found at:
Exponents and Roots Practice Problems: GED Math
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From GED Math. Copyright © 2008 by LearningExpress, LLC. All Rights Reserved.
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