# 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.

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

• $7^{2}$
 $7\times 7$ Seven to the second power, or seven squared, means seven times seven. $49$ Seven times seven is forty-nine.
Seven to the second power equals forty-nine.

• What is the area of a square with a side of 3 meters length?
 Area = (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. (3 meters)2 The length of the side is 3 meters, so the area is (3 meters) squared. $3\times 3$ meters2 3 squared is the same as 3 times 3. 9 square meters Three times three is nine.
So, the area of a square with a side length of 3 meters is 9 square meters.

• $c^{2}$  where $c=6$
 $6^{2}$ First, we replace the variable "c" in the expression with 6, which is what it equals. $6\times 6$ 6 squared equals 6 times 6. $36$ 6 times 6 equals 36.
So, c squared is 36.

• $x^{3}$  where $x=10$ .
 $10^{3}$ First, we replace the variable "x" in the expression with 10, which is what it equals. $10\times 10\times 10$ 10 to the third power, or 10 cubed, is equal to 10 times 10 times 10. $100\times 10$ 10 times 10 equals 100. $1000$ 100 times 10 equals 1000.
So, x to the third power is 1000.

• $y^{4}$  where $y=2$
 $2^{4}$ First, we replace the variable "y" in the expression with 2, which is what it equals. $2\times 2\times 2\times 2$ 2 to the fourth power is equal to 2 times 2 times 2 times 2. $4\times 2\times 2$ 2 times 2 equals 4. $8\times 2$ 4 times 2 equals 8. $16$ And 8 times 2 equals 16.
So, y to the fourth is 16.

• $3^{-3}$
 ${\frac {1}{3^{3}}}$ Three to the negative third power, which can be expressed as 1 over three cubed. ${\frac {1}{27}}$ Three cubed equals 3 times 3 times 3 which equals 27.
So, three to the negative third power equals one twenty-seventh.

## Practice Problems

Use / as the fraction line!

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

 $7^{2}=$ 11

 $12^{2}=$ 12

 $2^{4}=$ « Basic AlgebraExponents and Powers » Simple Operations Order of Operations