Solving Equations With Exponents

Instructor: David Karsner
Many equations have exponents. To solve these equations, you need to know how to deal with these exponents so that you can determine the value of the variable.

What are Exponents?

Let's start with a simple question. What is 5 x 5? I bet you said 25. Now what is 5 x 5 x 5? You might have used a calculator, but you probably got 125. Let's step things up a bit. What is 5 x 5 x 5 x 5 x 5 x 5 x 5 x 5 x 5 x 5, that's right 5 multiplied 10 times. You could use a calculator and type all those fives in; however, that's a lot of typing and lots of room to make errors. It would be nice to have a shortcut, and we do. Exponents are that shortcut. You can rewrite 5 multiplied ten times as 510, which will go nicely into the calculator. This lesson will explain three common scenarios with exponents and how to solve them.

The Basics

The primary vocabulary words to know for this lesson are base and exponent. The base is the number that is going to be multiplied. It is lower on the page than the exponent. The exponent is the number of times that the base will be multiplied. 721 means that seven (the base) will be multiplied 21 (the exponent) number of times.

Base and Exponent

When the Base is a Number

Many times you will see a number raised to an exponent equaling a variable. (Example: 25 = x). In this scenario, the variable is already isolated (by itself on one side of the equation). All you need to do is simplify the other side of the equation. (25 = 2 x 2 x 2 x 2 x 2 = 32). On scientific calculators, you can enter the base (2 in this case) select the 'raised to' button (it often looks like ^) and enter the exponent (5 in this case).

When the Base is a Variable

Many times you will see a variable raised to an exponent equaling a number. (Example x3 = 216). In this scenario, the x (the variable) is not isolated. You will need to solve for x (get x by itself on one side of the equation). To solve for x, you will need to apply the inverse operation, which will undo what has been done. For example, if you have added two, the inverse operation would be to subtract two. The inverse operation of raising x to a power (exponent) is to take the root of both sides of the equation. In our example, x has been raised to the third power (also known as taking the cube). The inverse operation of raising to the third power is taking the cube root of both sides (the cube root of x3 is x, the cube root of 216 is 6). Solving this equation gives us x=6. In most instances, finding the root of a number is most easily done on the calculator. Inspect your calculator to find where the root buttons are located. There are a variety of ways that calculators have included this button.

When the Exponent is a Negative

Many times you will find a number raised to a negative exponent equaling a variable (Example 3-2 =x). In this scenario, you should convert the negative exponent into a positive one before you simplify. To convert a negative exponent to a positive one, move the base and the exponent to the denominator of a fraction and make it positive. You will simplify from there.



Example 1


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