The Use of Exponents and Roots

Exponents are simply multiplying something by itself the given number of times.

For example:

x3 = x * x * x                                        r6 = r * r * r * r * r * r

23 = 2 * 2 * 2                                      74 = 7 * 7 * 7 * 7

            4 * 2                                                 49 * 7 * 7

                8                                                      49 * 49

                                                                           2401

Sometimes, an exponent is written with a ^. For example, 22 = 2^2 =4.

                                                                                            35 = 3^5 = 3*3*3*3*3 = 243

Remember anything to the first power (x1) is equal to itself (x1 = x)

and anything raised to the zero power is equal to 1 (x0 = 1).

A number with a negative power will result in a decimal answer.

Take an interactive practice quiz over basic use of exponents. 

Another type of exponents you will need to be able to use are the powers of ten. These are super easy once you understand how they work. The power that ten is being raised to will always be equal to the number of zeros behind the 1. But be careful, TEN is the only number this will work for.

For example:

101 = 10                102 = 100                103 = 1,000                104 = 10,000                105 = 100,000                106 = 1,000,000

With negative powers, you move the decimal point to the left the amount of digits as the power. For example,

10-4 = .0010 because you moved the decimal exactly four places to the left.

You will use a lot of powers of ten when you are asked to use scientific notation. Notice above that if ten is raised to a certain power the zeros in the answer are equal to the power ten was raised to.

Click here to review scientific notation

As you move on to a more advanced level of mathematics you may find that the use of exponents becomes more complicated.

You may, for example, be asked to take a problem with a variable in it  that is contained in parenthesis, like (x + 2), to a power.

Such as (x + 2)2. This is when it starts to get a little bit tricky, because merely distributing the exponents to the numbers/variables within the parenthesis is WRONG when the numbers within the parenthesis are being added or subtracted. You can, however, distribute the exponent if the numbers with in the parenthesis are ALL being multiplied or divided.

Properties of Exponents

Let a and b be real numbers, variables, or algebraic expressions, and let m and n be integers. (All denominators and bases are nonzero).

Standard Property                            

1. am + an = a(m+n)

2. am / an = a(m-n)

3. a-n = 1/an = (1/a)n 

4. a0 = 1

5. (ab)m = ambm

6. (am)n = amn

7. (a/b)m= am / bm

8. |a2| = |a|2 = a2            Absolute Values (true for all even powers, but not odd ones (-33 = -27))

When dealing with addition and subtraction you will need to be able to "F.O.I.L.."  Because (x + 2) ^ 2 = (x + 2) * (x + 2)    F.O.I.L stands for First. Inside. Outside. Last. respectively.  In order to F.O.I.L this properly, you would first multiply x*x, then x*2, then the other x*2, and finally 2*2. Your answer should be something like x2+2x+2x+4 which then can be reduced to x2+4x+4. You can also use the box method for FOILing. 

Click review how to "F.O.I.L."

When dealing with multiplication and division ONLY (it will not work if there is any addition or subtraction involved) you will need to be able to use the rules of powers. 

Click here to review the rules of powers

Roots, also called radicals, are the opposite operation of an exponent. They are a way to undo exponents. To show you are doing a root operation, a radical sign is used (√)

To undo x2, you square root it.

To undo a power other than 2, such as 4, you take the 4th root of that number. You can also take the number to the 1/4 power, which is shown as follows:

∜16 = 16^1/4= 2

If you are doing both a root and an exponent operation to the same number, you can change in to just an exponential form. The root would be the denominator of the exponent, and the exponent would be the numerator of the exponent. For example, the 3rd root of ten to the fourth would look as follows:

∛(10) ^4 = 10^4/3= 21.54