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

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Algebra is the most boring thing you'll ever do


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: 5^2=25

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


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[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.


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 squared equals 6 times 6.


6 times 6 equals 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 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.


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 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.


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]

Practice Problems[edit]

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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Exponents and Powers
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