Intro to Exponents

 Here's what an exponent and a base look like:

$\blueD4^\goldD3$start color #11accd, 4, end color #11accd, start superscript, start color #e07d10, 3, end color #e07d10, end superscript

The small number written above and to the right of a number is called an $\goldD{\text{exponent}}$start color #e07d10, start text, e, x, p, o, n, e, n, t, end text, end color #e07d10. The number underneath the exponent is called the $\blueD{\text{base}}$start color #11accd, start text, b, a, s, e, end text, end color #11accd. In this example, the base is $\blueD4$start color #11accd, 4, end color #11accd, and the exponent is $\goldD3$start color #e07d10, 3, end color #e07d10.

Here's an example where the base is $\blueD7$start color #11accd, 7, end color #11accd, and the exponent is $\goldD5$start color #e07d10, 5, end color #e07d10:

$\blueD7^\goldD5$start color #11accd, 7, end color #11accd, start superscript, start color #e07d10, 5, end color #e07d10, end superscript

An exponent tells us to multiply the base by itself that number of times. In our example, $\blueD4^\goldD3$start color #11accd, 4, end color #11accd, start superscript, start color #e07d10, 3, end color #e07d10, end superscript tells us to multiply the base of $\blueD4$start color #11accd, 4, end color #11accd by itself $\goldD3$start color #e07d10, 3, end color #e07d10 times:

$\blueD4^\goldD3 =\blueD4 \times \blueD4 \times \blueD4$start color #11accd, 4, end color #11accd, start superscript, start color #e07d10, 3, end color #e07d10, end superscript, equals, start color #11accd, 4, end color #11accd, times, start color #11accd, 4, end color #11accd, times, start color #11accd, 4, end color #11accd

Once we write out the multiplication problem, we can easily evaluate the expression. Let's do this for the example we've been working with:

$\blueD4^\goldD3 =\blueD4 \times \blueD4 \times \blueD4$start color #11accd, 4, end color #11accd, start superscript, start color #e07d10, 3, end color #e07d10, end superscript, equals, start color #11accd, 4, end color #11accd, times, start color #11accd, 4, end color #11accd, times, start color #11accd, 4, end color #11accd

$\phantom{\blueD4^\goldD3}= 16 \times 4$empty space, equals, 16, times, 4

$\phantom{\blueD4^\goldD3}= 64$empty space, equals, 64

The main reason we use exponents is because it's a shorter way to write out big numbers. For example, let's say we want to express the following:

$\blueD2 \times \blueD2 \times \blueD2 \times \blueD2 \times \blueD2 \times \blueD2$start color #11accd, 2, end color #11accd, times, start color #11accd, 2, end color #11accd, times, start color #11accd, 2, end color #11accd, times, start color #11accd, 2, end color #11accd, times, start color #11accd, 2, end color #11accd, times, start color #11accd, 2, end color #11accd

That's really long to write. My hands hurt just from typing it! Instead we can see that $\blueD2$start color #11accd, 2, end color #11accd is multiplied by itself $\goldD6$start color #e07d10, 6, end color #e07d10 times. This means we can write the same thing with $\blueD2$start color #11accd, 2, end color #11accd as the base and $\goldD6$start color #e07d10, 6, end color #e07d10 as the exponent:

$\blueD2 \times \blueD2 \times \blueD2 \times \blueD2 \times \blueD2 \times \blueD2 = \blueD2^\goldD6$start color #11accd, 2, end color #11accd, times, start color #11accd, 2, end color #11accd, times, start color #11accd, 2, end color #11accd, times, start color #11accd, 2, end color #11accd, times, start color #11accd, 2, end color #11accd, times, start color #11accd, 2, end color #11accd, equals, start color #11accd, 2, end color #11accd, start superscript, start color #e07d10, 6, end color #e07d10, end superscript

Cool, lets make sure we understand exponents by trying some practice problems.

Practice set:

PROBLEM 1G

Evaluate.

$3^3 =$3, cubed, equals 

Correct

Challenge set:

PROBLEM 1D

Is $2^{10}$2, start superscript, 10, end superscript or $10^2$10, squared larger?

Choose 1 answer:

Choose 1 answer:

• (Choice A, Checked, Correct)

  CORRECT (SELECTED)

  $2^{10}$2, start superscript, 10, end superscript is larger than $10^2$10, squared
• (Choice B)

  B

  $10^2$10, squared is larger than $2^{10}$2, start superscript, 10, end superscript

Correct

It's easy to evaluate $10^2$10, squared:

$10^2 = 10 \times 10 = 100$10, squared, equals, 10, times, 10, equals, 100

So what we're really wondering is how $100$100 compares to $2^{10}$2, start superscript, 10, end superscript.

Let's write out $2^{10}$2, start superscript, 10, end superscript:

$2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2$2, times, 2, times, 2, times, 2, times, 2, times, 2, times, 2, times, 2, times, 2, times, 2

If we multiply this expression out, we get $1024$1024.

So, $2^{10}$2, start superscript, 10, end superscript is larger than $10^2$10, squared because $1024$1024 is larger than $100$100.

This goes to show how fast numbers can increase when we use exponents-- this is called exponential growth. We'll learn more about exponential growth in algebra.