# 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: ${\displaystyle 5^{2}=25}$

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.

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

${\displaystyle 2^{0}=1}$
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:

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

## Example Problems

Let's evaluate these expressions.

• ${\displaystyle 7^{2}}$
 ${\displaystyle 7\times 7}$ Seven to the second power, or seven squared, means seven times seven. ${\displaystyle 49}$ 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. ${\displaystyle 3\times 3}$  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.

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

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

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

• ${\displaystyle 3^{-3}}$
 ${\displaystyle {\frac {1}{3^{3}}}}$ Three to the negative third power, which can be expressed as 1 over three cubed. ${\displaystyle {\frac {1}{27}}}$ 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

 ${\displaystyle 6^{2}=}$

2

 ${\displaystyle 2^{3}=}$

3

 ${\displaystyle 4^{2}=}$

4

 ${\displaystyle 5^{3}=}$

5

 ${\displaystyle 2^{4}=}$

6

 ${\displaystyle 9^{2}=}$

7

 ${\displaystyle 8^{2}=}$

8

 ${\displaystyle 5^{-3}=}$

9

 ${\displaystyle 6^{0}=}$

10

 ${\displaystyle 7^{2}=}$

11

 ${\displaystyle 12^{2}=}$

12

 ${\displaystyle 2^{4}=}$

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