# Basic Algebra/Introduction to Basic Algebra Ideas/Exponents and Powers

## Vocabulary

Exponent
a number written in superscript that denotes how many times the base will be multiplied by itself.
the number to be multiplied by itself.

Example: $5^2=25$

In this example, the base is 5 and the exponent is 2.

## Lesson

We use exponents to show when we're multiplying the same number more than one time.

$3 \cdot 3 = 3^{2}$
Three times three equals three to the second power (or three squared)
$3 \cdot 3 \cdot 3 = 3^{3}$
Three times three times three equals three to the third power (or three cubed)
$3\cdot3\cdot3\cdot3= 3^{4}$
Three times three times three times three equal three to the fourth power
$2\cdot2\cdot2 = 2^{3}$
Two times two times two equals two to the third power

Note that any nonzero number raised to the 0 power is always equal to 1.

$2^{0} = 1$
Two to the zero power equals one

We can also raise any number to a negative exponent. This is called the inverse exponent and places the number on the bottom of a fraction with a 1 on top:

$2^{-2} = \frac{1}{2^{2}} = \frac{1}{4}$
Two to the negative two equals one over two to the second power

## Example Problems

Let's evaluate these expressions.

Example 1

$7^{2}$

Seven to the second power, or seven squared, means seven times seven.

$7\cdot7$

Seven times seven is forty-nine.

49

Seven to the second power equals forty-nine.
Example 2

Area of a square = (length of the side) ^2

The area, or space inside, of a square is equal to the length of the side of the square to the second power.

Area of a square with side length 3 meters

If the square had a side length of 3 meters,

(3 meters)^2

Then the area would be (3 meters) squared.

$3\cdot3$ meters^2

3 squared is the same as 3 times 3.

9 square meters

So, the area of a square with a side length of 3 meters is 9 square meters.
Example 3

$c^{2}$ where c=6

First, we replace the variable "c" in the expression with 6, which is what it equals.

$6^{2}$

6 squared equals 6 times 6.

$6\cdot6$

6 times 6 equals 36.

36

So, c squared is 36.
Example 4

$x^{3}$ where x = 10.

First, we replace the variable "x" in the expression with 10, which is what it equals.

$10^{3}$

10 to the third power, or 10 cubed, is equal to 10 times 10 times 10.

$10\cdot10\cdot10$

10 times 10 equals 100.

$100\cdot10$

100 times 10 equals 1000.

1000

So, x to the third power is 1000.
Example 5

$y^{4}$ where y = 2

First, we replace the variable "y" in the expression with 2, which is what it equals.

$2^{4}$

2 to the fourth power is equal to 2 times 2 times 2 times 2.

$2\cdot2\cdot2\cdot2$

2 times 2 equals 4.

$4\cdot2\cdot2$

4 times 2 equals 8.

$8\cdot2$

And 8 times 2 equals 16.

16

So, y to the fourth is 16.
Example 6

$3^{-3}$

Three to the negative third power, which can be expressed as 1 over three cubed.

$\frac{1}{3^{3}}$

Three cubed equals 3 times 3 times 3 which equals 27.

$\frac{1}{27}$

So, three to the negative third power equals one twenty-seventh.

## Practice Problems

Evaluate the following expressions:

1. $6^{2}$
2. $2^{3}$
3. $4^{2}$
4. $5^{3}$
5. $2^{4}$
6. $9^{2}$
7. $8^{2}$
8. $5^{-3}$
9. $6^{0}$
10. $2^{4}$
 « Basic Algebra Exponents and Powers » Simple Operations Order of Operations