# Intro to exponents

Learn how to use exponents and bases. For example, writing 4 x 4 x 4 x 4 x 4 with an exponent.

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

$\blueD4^\goldD3$start color blueD, 4, end color blueD, start superscript, start color goldD, 3, end color goldD, end superscript

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

Here's an example where the base is $\blueD7$start color blueD, 7, end color blueD, and the exponent is $\goldD5$start color goldD, 5, end color goldD:

$\blueD7^\goldD5$start color blueD, 7, end color blueD, start superscript, start color goldD, 5, end color goldD, end superscript

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

$\blueD4^\goldD3 =\blueD4 \times \blueD4 \times \blueD4$start color blueD, 4, end color blueD, start superscript, start color goldD, 3, end color goldD, end superscript, equals, start color blueD, 4, end color blueD, times, start color blueD, 4, end color blueD, times, start color blueD, 4, end color blueD

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 blueD, 4, end color blueD, start superscript, start color goldD, 3, end color goldD, end superscript, equals, start color blueD, 4, end color blueD, times, start color blueD, 4, end color blueD, times, start color blueD, 4, end color blueD$\phantom{\blueD4^\goldD3}= 16 \times 4$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 blueD, 2, end color blueD, times, start color blueD, 2, end color blueD, times, start color blueD, 2, end color blueD, times, start color blueD, 2, end color blueD, times, start color blueD, 2, end color blueD, times, start color blueD, 2, end color blueD

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

$\blueD2 \times \blueD2 \times \blueD2 \times \blueD2 \times \blueD2 \times \blueD2 = \blueD2^\goldD6$start color blueD, 2, end color blueD, times, start color blueD, 2, end color blueD, times, start color blueD, 2, end color blueD, times, start color blueD, 2, end color blueD, times, start color blueD, 2, end color blueD, times, start color blueD, 2, end color blueD, equals, start color blueD, 2, end color blueD, start superscript, start color goldD, 6, end color goldD, end superscript

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

# Practice

**Write $7 \times 7 \times 7$7, times, 7, times, 7 using an exponent.**

This is $\blueD7$start color blueD, 7, end color blueD multiplied by itself $\goldD3$start color goldD, 3, end color goldD times:

$\blueD7^\goldD3$start color blueD, 7, end color blueD, start superscript, start color goldD, 3, end color goldD, end superscript

**Write $5 \times 5 \times 5 \times 5 \times 5 \times 5$5, times, 5, times, 5, times, 5, times, 5, times, 5 using an exponent.**

This is $\blueD5$start color blueD, 5, end color blueD multiplied by itself $\goldD6$start color goldD, 6, end color goldD times:

$\blueD5^\goldD6$start color blueD, 5, end color blueD, start superscript, start color goldD, 6, end color goldD, end superscript

**Write $10 \times 10 \times 10 \times 10$10, times, 10, times, 10, times, 10 using an exponent.**

This is $\blueD{10}$start color blueD, 10, end color blueD multiplied by itself $\goldD4$start color goldD, 4, end color goldD times:

$\blueD{10}^\goldD4$start color blueD, 10, end color blueD, start superscript, start color goldD, 4, end color goldD, end superscript

**Evaluate.**

$5^2 =$5, start superscript, 2, end superscript, equals

**Your answer should be**- an integer, like $6$6
- a
*simplified proper*fraction, like $3/5$3, slash, 5 - a
*simplified improper*fraction, like $7/4$7, slash, 4 - a mixed number, like $1\ 3/4$1, space, 3, slash, 4
- an
*exact*decimal, like $0.75$0, point, 75 - a multiple of pi, like $12\ \text{pi}$12, space, p, i or $2/3\ \text{pi}$2, slash, 3, space, p, i

$\blueD5^\goldD2 =\blueD5 \times \blueD5$start color blueD, 5, end color blueD, start superscript, start color goldD, 2, end color goldD, end superscript, equals, start color blueD, 5, end color blueD, times, start color blueD, 5, end color blueD

$\phantom{\blueD4^\goldD3}= 25$empty space, equals, 25

**Evaluate.**

$2^3 =$2, start superscript, 3, end superscript, equals

**Your answer should be**- an integer, like $6$6
- a
*simplified proper*fraction, like $3/5$3, slash, 5 - a
*simplified improper*fraction, like $7/4$7, slash, 4 - a mixed number, like $1\ 3/4$1, space, 3, slash, 4
- an
*exact*decimal, like $0.75$0, point, 75 - a multiple of pi, like $12\ \text{pi}$12, space, p, i or $2/3\ \text{pi}$2, slash, 3, space, p, i

$\blueD2^\goldD3 =\blueD2 \times \blueD2 \times \blueD2$start color blueD, 2, end color blueD, start superscript, start color goldD, 3, end color goldD, end superscript, equals, start color blueD, 2, end color blueD, times, start color blueD, 2, end color blueD, times, start color blueD, 2, end color blueD

$\phantom{\blueD4^\goldD3}= 4 \times 2$empty space, equals, 4, times, 2

$\phantom{\blueD4^\goldD3}= 8$empty space, equals, 8

**Evaluate.**

$1^4 =$1, start superscript, 4, end superscript, equals

**Your answer should be**- an integer, like $6$6
- a
*simplified proper*fraction, like $3/5$3, slash, 5 - a
*simplified improper*fraction, like $7/4$7, slash, 4 - a mixed number, like $1\ 3/4$1, space, 3, slash, 4
- an
*exact*decimal, like $0.75$0, point, 75 - a multiple of pi, like $12\ \text{pi}$12, space, p, i or $2/3\ \text{pi}$2, slash, 3, space, p, i

$\blueD1^\goldD4 =\blueD1 \times \blueD1 \times \blueD1 \times \blueD1$start color blueD, 1, end color blueD, start superscript, start color goldD, 4, end color goldD, end superscript, equals, start color blueD, 1, end color blueD, times, start color blueD, 1, end color blueD, times, start color blueD, 1, end color blueD, times, start color blueD, 1, end color blueD

$\phantom{\blueD4^\goldD3}= 1$empty space, equals, 1

# Challenge problem

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

**Why do you think this is?**

It's easy to evaluate $10^2$10, start superscript, 2, end superscript:

$10^2 = 10 \times 10 = 100$10, start superscript, 2, end superscript, 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, start superscript, 2, end superscript 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.