Last modified on 23 March 2014, at 20:37

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

VocabularyEdit

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

Example: 5^2=25

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

LessonEdit

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 ProblemsEdit

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 GamesEdit

Practice ProblemsEdit

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}
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