# 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.
the number to be multiplied by itself.

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

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.

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

Let's evaluate these expressions.

Example 1

${\displaystyle 7^{2}}$

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

${\displaystyle 7\cdot 7}$

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.

${\displaystyle 3\cdot 3}$  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

${\displaystyle c^{2}}$  where c=6

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

${\displaystyle 6^{2}}$

6 squared equals 6 times 6.

${\displaystyle 6\cdot 6}$

6 times 6 equals 36.

36

So, c squared is 36.
Example 4

${\displaystyle x^{3}}$  where x = 10.

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

${\displaystyle 10^{3}}$

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

${\displaystyle 10\cdot 10\cdot 10}$

10 times 10 equals 100.

${\displaystyle 100\cdot 10}$

100 times 10 equals 1000.

1000

So, x to the third power is 1000.
Example 5

${\displaystyle y^{4}}$  where y = 2

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

${\displaystyle 2^{4}}$

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

${\displaystyle 2\cdot 2\cdot 2\cdot 2}$

2 times 2 equals 4.

${\displaystyle 4\cdot 2\cdot 2}$

4 times 2 equals 8.

${\displaystyle 8\cdot 2}$

And 8 times 2 equals 16.

16

So, y to the fourth is 16.
Example 6

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

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

${\displaystyle {\frac {1}{3^{3}}}}$

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

${\displaystyle {\frac {1}{27}}}$

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

## Practice ProblemsEdit

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. ${\displaystyle 8^{2}}$  12344
8. ${\displaystyle 5^{-3}}$
9. ${\displaystyle 6^{0}}$
10. ${\displaystyle 2^{4}}$
Solution
1. ${\displaystyle 6^{2}}$  36
2. ${\displaystyle 2^{3}}$ 8
3. ${\displaystyle 4^{2}}$  16
4. ${\displaystyle 5^{3}}$  125
5. ${\displaystyle 2^{4}}$  16
6. ${\displaystyle 9^{2}}$  81
7. ${\displaystyle 8^{2}}$  64
8. ${\displaystyle 5^{-3}}$  1/125
9. ${\displaystyle 6^{0}}$  1
10. ${\displaystyle 2^{4}}$ 16
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