Math 10: Week 2 – Exponent Rules

What is an exponent?

An exponent refers to the number of times a number is multiplied by itself. For example, 2 to the 3rd (written like this: 2³) means:

2 x 2 x 2 = 8.

2³ is not the same as 2 x 3 = 6.

Remember that a number raised to the power of 1 is itself. For example,

a¹ = a

5¹ = 5.

There are some special cases:

1. a⁰ = 1

When an exponent is zero, as in 6⁰, the expression is always equal to 1.

a⁰ = 1

6⁰ = 1

14,356⁰ = 1

You can have a variable to a given power, such as a³, which would mean a x a x a. 

The Rules.

There are five rules for working with exponents:

1. a^m * aⁿ = a^(m+n)

2. (a * b)ⁿ = aⁿ * bⁿ

3. (a^m)ⁿ = a^{(m * n)}

4. a^m / aⁿ = a^(m-n)

5. (a/b)ⁿ = aⁿ / bⁿ

a^m * =aⁿ  a^(m+n) says that when you take a number, a, multiplied by itself m times, and multiply that by the same number a multiplied by itself n times, it’s the same as taking that number a and raising it to a power equal to the sum of m + n.

Here’s an example where

a = 3
m = 4
n = 5

3⁴ * 3⁵ = 3⁽⁴⁺⁵⁾ = 3⁹ = 19,683

(a * b)ⁿ = aⁿ * bⁿ says that when you multiply two numbers, and then multiply that product by itself n times, it’s the same as multiplying the first number by itself n times and multiplying that by the second number multiplied by itself n times.

Let’s work out an example where

a = 3
b = 6
n = 5

(3 * 6)⁵ = 3⁵ * 6⁵

18⁵ = 3⁵ * 6⁵ = 243 * 7,776 = 1,889,568

(a^m)ⁿ = a^{(m * n)} says that when you take a number, a , and multiply it by itself m times, then multiply that product by itself n times, it’s the same as multiplying the number a by itself m * n times.

Let’s work out an example where

a = 3
m = 4
n = 5

(3⁴)⁵ = 3^{(4 * 5)} = 3²⁰ = 3,486,784,401

a^m / aⁿ = a^(m-n) says that when you take a number, a, and multiply it by itself m times, then divide that product by a multiplied by itself n times, it’s the same as a multiplied by itself m–n times.

Here’s an example where

a = 3
m = 4
n = 5

3⁴ / 3⁵ = 3^(4-5) = 3^-1 (Remember how to raise a number to a negative exponent.)

3⁴ / 3⁵ = 1 / 3¹ = 1/3

(a/b)ⁿ = aⁿ / bⁿ says that when you divide a number, a by another number, b, and then multiply that quotient by itself n times, it is the same as multiplying the number by itself n times and then dividing that product by the number b multiplied by itself n times.

Let’s work out an example where

a = 3
m = 4
n = 5

(3/6)⁵ = 3⁵ / 6⁵

Remember 3/6 can be reduced to 1/2. So we have:

(1/2)⁵ = 243 / 7,776 = 0.03125

 

 

 

 

sources

1. http://www.mclph.umn.edu/mathrefresh/exponents.html