Exploring Exponents: Exponent Rules Educational Resources K12 Learning, Expressions and Equations, Pre-Algebra, Rules and Properties, Math Lesson Plans, Activities, Experiments, Homeschool Help

Lesson Plan - Get It!

Audio:

What do you do when you have multiple exponents in an equation?
Do you turn and run or tackle it with ease?

With the help of a few exponent rules, you are sure to do the second!

It's true, exponents are powerful, but that doesn't mean you need to be afraid.

Just like most things in math, there are specific properties and rules that you can follow to help you solve such equations.

Let's start with a quick review of exponents.

The exponent of a number says how many times to multiply a number times itself.

exponent

Ways to read it:

two to the fourth power
two to the power of four

It simply means two times itself 4 times:

2⁴ = 2 x 2 x 2 x 2
2⁴ = 16

Remember, you take the base times ITSELF, not the base times the exponent! This is a very easy (and common) mistake to make, so be careful.

If you need more time to review exponents, take a look at the Elephango lesson found in Additional Resources in the right-hand sidebar.

Now, what about when you stumble across a problem that has multiple exponents that are being multiplied, or maybe terms with exponents that are being divided?

Do you turn around and just run?

No way! There are very specific rules you can follow when you see multiple exponents together in a problem.

Take a minute to make a foldable chart to help you solve problems today and in the future!

Grab a piece of paper and fold it into 3 columns.
You will need 9 rows, so add 8 folds for these rows.
Open it up and copy this chart onto it:

Rule	Example	What It Means
x¹ = x	2¹ = 2	Any number to the 1^st power is just itself.
x⁰ = 1	2⁰ = 1	Any number to the 0 power is 1.
(x^m)(xⁿ) = x^m+n	(x⁶)(x²) = x⁶⁺² = x⁸	When you multiply two terms with the same base, you keep the base the same and just add the exponents.
(x^m)ⁿ = x^(m)(n)	(x⁶)² = x⁽⁶⁾⁽²⁾ = x¹²	If an exponent is raised to another power, you keep the base and multiply the exponent and power together.
(xy)^m = x^my^m	(2xy)² = 2²x²y² = 4x²y²	If you have a product inside parentheses and a power outside the parentheses, that power will go on each element inside the parenthesis. (Think of distributing the power.)
x^m/xⁿ= x^m-n	x⁶/x² = x^6-2 = x⁴	When you divide two terms with the same base, you keep the base the same and subtract the exponents.
(x/y)^m = x^m/y^m	(2x/y)⁶ = 2⁶x⁶/y⁶ = 64x⁶/y⁶	If you have a quotient inside parentheses and a power outside the parentheses, that power will go on each element inside the parentheses. (Think of distributing the power.)
x^-m = 1/x^m	3x^-4 = 3/x⁴	When you see a negative exponent, flip the base to the bottom of a fraction and keep the positive version of the exponent with it.

Here are a few more examples of these rules for you to look over:

What is (4b)³?

= 4³b³

=64b³

What is 1/4 of 4⁸?

	=	1	x	4⁸	=	4⁸
		4				4

= 4^8-1

= 4⁷ = 16,384

	Simplify:	18x⁶y⁴
		3x⁵y²

= (18 ÷ 3) (x^6-5)(y^4-2)

= 6xy²

Notice how, in the last example, you look at each term separately. You divide the numbers normally, then follow the rules and subtract the exponents for each of the variables.

Please watch this Exponents Rules Part 1 | Exponents, radicals, and scientific notation | Pre-Algebra | Khan Academy video to see examples using these exponents rules:

If you feel like your head is spinning with all these rules, remember: you only use one at a time. You can use the chart above to help you out as you work through some problems in this lesson and at any time.

Go to the Got It? section now for some practice using these rules.