You'll often find exponents within equations and calculations being added, subtracted, divided, multiplied, etc. In these cases, you need to know how to deal with exponents and how these equations will affects the exponents.
These are called the "rules of exponents." Here are the most common ones you need to know.
When you're adding numbers with exponents, you take the exponent of the bases first and then add the results of those products. Here's a step-by-step example:
5(2) + 6(3)
= (5*5) + (6*6*6)
= 25 + 216
If you are adding exponents that have the same base number, you can simplify it like this:
Just as with adding, when you're subtracting numbers with exponents, you take the exponent of the base numbers first and then subtract the results of those products. For example:
7(4) - 3(5)
= (7*7*7*7) - (3*3*3*3*3)
= 2,401 - 243
Sometimes, you'll see exponents with the same base number multiplied. In this case, you add together the exponents. This is called the product rule.
This looks like this:
Here's an example problem:
10(3) * 10(2)
You might also find a situation where bases being multiplied are taken to the same exponent like this:
(a * c)n
In this case, you would apply the exponent to each of the bases individually then multiply the results of those products, like this:
= a(n) * c(n)
Dividing two exponents that share the same base can be simplified by subtracting the exponents. This is called the quotient rule.
Here it is visually so you can understand it a bit better:
The only stipulation here is that the base cannot be equal to zero (you cannot divide by 0).
Similar to the multiplication rule, if you have two bases being divided that are taken to the power of n, you apply that n exponent to both bases, then divide. This looks like this:
(a / c)n
= (a(n)) / (c(n))
The Exponent of an Exponent
Taking an exponent to the power of another exponent can be simplified by multiplying the two exponents together. This is called the power rule, and it looks like this:
Here's an example:
When you have an exponent that's negative, this means you get the reciprocal of the base raised to that power. This is a bit wordy, so let's look at what this looks like:
However, in this case, the base cannot be 0. This would result in dividing by 0, which cannot be done.
If you have an exponent that's a fraction that has a numerator of 1 and a denominator of "n", you take the nth root of the base number.
For example, if you had 16(1/2), you would take the square root of 16 to get an answer of 4.
If you had 125(1/3), you would take the cube root of 125, which would give you an answer of 5.
Exponent as 1 or 0
Any integer to the power of 1 will simply give you that integer.
101 = 10
1671 = 167
01 = 0.
Any integer taken to the 0th power will give you an answer of 1.