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.
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.

Lesson

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

$3\times 3=3^{2}$: Three times three equals three to the second power (or three squared)

$3\times 3\times 3=3^{3}$: Three times three times three equals three to the third power (or three cubed)

$3\times 3\times 3\times 3=3^{4}$: Three times three times three times three equal three to the fourth power

$2\times 2\times 2=2^{3}$: 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.

$2^{0}=1$: 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

Let's evaluate these expressions.

$7^{2}$

$7\times 7$	Seven to the second power, or seven squared, means seven times seven.
$49$	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.
$3\times 3$ 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.

$c^{2}$ where $c=6$

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

So, c squared is 36.

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

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

So, x to the third power is 1000.

$y^{4}$ where $y=2$

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

So, y to the fourth is 16.

$3^{-3}$

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