Section 3: Exponents and Radicals

Positive Exponents

Exponents (or powers) indicate that a number is multiplied by itself a certain number of times. In other words, exponents are just a short hand way of writing out a number multiplied by itself.

For example, if we see 

Image of what 5 squared

(said “five squared” and sometimes written as 5^2), it is the same thing as 5 * 5. The number that is to be multiplied by itself is called the base. How many times the number is to be multiplied by itself is called the exponent or the power.

Be very careful when a negative number is raised to a power. Be sure to keep track of the sign. A negative number multiplied by itself an even number of times will give a positive number result. A negative number multiplied by itself an odd number of times will give a negative number.

For example

Image of what -2 squared looks like

 is equal to (-2)*(-2) = 4. (try it!) Note that the exponent is even and the resulting answer is positive. However, 

 is equal to (-2)*(-2)*(-2) = -8. (try it!) Note that the exponent in this case is odd and the resulting answer is negative.

Image of what -2 cubed looks like

Some calculators have a “^” key that allows you to enter in an expression with an exponent directly. In other words, you can key in 3^2= to get 9. See below.

Image of a calculator with the “^” key highlighted

If your calculator does not have a “^” key, then you need to type in the multiplication expression—i.e. for 

Image of what 3 squared looks like

you need to type 3 * 3.

Example: Find

Image of what 7.865 squared looks like

7.8652 = 7.865 * 7.865 = 61.858

Example: Find 

Image of what -3.5 to the 5th power looks like

= (-3.5)*(-3.5)*(-3.5)*(-3.5)*(-3.5) = -525.22

Negative Exponents

Exponents can also be negative. Negative exponents are treated differently. We will not be using negative exponents in the class.

Radicals

Radicals (or roots) are sort of the opposite of positive exponents. The radical is the symbol placed over a number. The most common radical is the square root, 

Image of what a square root symbol looks like

The square root is the number you need to multiply by itself twice to get the number that is under the radical. For example, the square root of 4, or 

Image of what the square root of 4 looks like

, is equal to 2 (or -2) because when you multiply 2 (or -2) by itself two times, you get 4.

There are other higher order roots, such as the cube root, 

Image of what a cube root looks like

, or the fourth root, 

Image of what a fourth root looks like

 In these cases, the root is the number that is multiplied by itself 3 or 4 times to get the number under the radical. For example, the cube root of 8,

Image of what the cube root of 8 looks like

, is 2 because 2*2*2 or 23 equals 8.

Your calculator should have a square root button. On some calculators, you need to hit the square root button first and then the number. On other calculators, you hit the buttons in reverse order. Note: Your calculator will only show you the positive square root. Graphing calculators will have options for calculating higher order roots.

Image of a calculator with the square root symbol circled

Example: Find 

Image of what the square root of 5 looks like

Since 5*5 = 25, 5 is a square root of 25. (-5 is also a square root of 25.)

Take Exponents and Radicals Assessment
Return to the Math Tutorial