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

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## Vocabulary[edit]

- Exponent
- a number written in superscript that denotes how many times the base will be multiplied by itself.
- Base (or radix)
- the number to be multiplied by itself.

Example:

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

## Lesson[edit]

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

- Three times three equals three to the second power (or three squared)

- Three times three times three equals three to the third power (or three cubed)

- Three times three times three times three equal three to the fourth power

- 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.

- 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:

- Two to the negative two equals one over two to the second power

## Example Problems[edit]

Let's evaluate these expressions.

- Example 1

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

- 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.

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

where c=6

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

- 6 squared equals 6 times 6.

- 6 times 6 equals 36.

36

- So, c squared is 36.

- Example 4

where x = 10.

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

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

- 10 times 10 equals 100.

- 100 times 10 equals 1000.

1000

- So, x to the third power is 1000.

- Example 5

where y = 2

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

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

- 2 times 2 equals 4.

- 4 times 2 equals 8.

- And 8 times 2 equals 16.

16

- So, y to the fourth is 16.

- Example 6

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

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

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

## Practice Games[edit]

- http://www.math.com/school/subject2/practice/S2U2L2/S2U2L2Pract.html
- http://www.quia.com/pop/50485.html (scientific notation)
- http://www.softschools.com/math/games/exponents_practice.jsp
- http://www.quia.com/quiz/358716.html (King Kong Scientific Notation)
- http://www.shodor.org/interactivate/activities/OrderOfOperationsFou/ (order of operations including exponents)

## Practice Problems[edit]

Evaluate the following expressions:

- 12344

Solution

- 36
- 8
- 16
- 125
- 16
- 81
- 64
- 1/125
- 1
- 16