Exponents

Exponents are often taught, and understood, procedurally. IF exponents were never to be seen in future mathematics, that would be okay.  But, exponents are not a one-off topic.  In fact, they’re a way of writing repeated multiplication, just like multiplication is a way of writing repeated addition.  Those basic operations are important to know and understand.

The materials you’ll find on this page are designed, through great pains and lots of trial and error, to offer an alternative approach.  There are no tricks or gimmicks, but it is very student friendly.

The approach is referenced to the mathematical facts about notation, repeated multiplication, and patterns we find in exponents.  It is all uncovered in a guided discovery-like approach.

On this page you’ll find everything about very basics about exponents.  The next page discusses all of the “rules,” and their origins.  The third page deals with scientific notation, which involves powers of ten, which are just exponents with a base of ten.  The final page explains rational exponents.   Use the tabs below to work your way through the materials.  If you’re a student, read the notes, watch the videos, try the practice problems.  If you have questions, please email us:  thebeardedmathman@gmail.com

If you’re a teacher and would like to have an organized, easy to use and reference zip file of all of the materials found on this page, and materials not published on this site, consider downloading the Exponents Packet.

exponents part 1

Exponents

Number Unit

Part 1

One of the biggest things to understand about math is how it is written.  The spatial arrangement of characters is syntax.  Syntax, in English, refers to the arrangements of words to convey meaning.

Exponents are just a way of writing repeated multiplication.  If we are multiplying a number by itself repeatedly, we can use an exponent to tell how many times the number is being multiplied.  That’s it.  Nothing tricky exists with exponents, no new operations or concepts to tackle.  If you’re familiar with multiplication and its properties, exponents should be accessible.

That said, it is not without its pitfalls.  A balance between conceptual understanding and procedural short-cuts is needed to avoid those pitfalls.  The only way to strike that balance is through a careful progression of exercises and examples.  An answer-getting mentality will lead to big troubles with exponents.  People wishing to learn how exponents work must seek understanding.

Let’s establish some facts that will come into play with this first part of exponents.

1.       Exponents are repeated multiplication

To simplify simple expressions with exponents you only need to know a few short-cuts, but to recall and understand, we need more.  These facts are important.

With an exponential expression we have a base, the number being multiplied by itself, and the exponent, the small number on the top right of the base which describes how many times the base is being multiplied by itself.

a5

The number a is the base.  We don’t know what a is other than it is a number.  It’s not a big deal that we don’t know exactly what number it is, w

Five is the exponent, which means there are five a’s, all multiplying together, like this: a·a·a·a·a.

Something to keep in mind is that this expression equals another number.  Since we don’t know what a is, we cannot find out exactly what it is, but we do know it’s a perfect 5th power number, like 32.   See, 25 = 32.

What if we had another number multiplying with a5, like this:

a5·b3

If we write this out, without the exponents we see we have 5 a’s and 3 b’s, all multiplying together.  We don’t know what a or b equals, but we do know they’re multiplying so we could change the order of multiplication (commutative property) or group them together anyway we wish (associative property) without changing the value.

a5·b3 = a· a· a· a· a· b· b· b

And these would be the same:

(aa)(aaa)(bbb)$\left(a\cdot a\right)\left(a\cdot a\cdot a\right)\left(b\cdot b\cdot b\right)$

(aa)[(aaa)(bbb)]$\left(a\cdot a\right)\left[\left(a\cdot a\cdot a\right)\left(b\cdot b\cdot b\right)\right]$

(aa)[ab]3$\left(a\cdot a\right){\left[ab\right]}^{3}$

a2[ab]3${a}^{2}{\left[ab\right]}^{3}$

This is true because the brackets group together the a and b, making them both the base.  The brackets put them together.  The base is ab, and the exponent is 3.  This means we have ab multiplied by itself three times.

Keep in mind, these are steps but exploring how exponents work to help you learn to read the math for the intended meaning behind the spatial arrangement of bases, parenthesis and exponents.

Now, the bracketed expression above is different than ab3, which is a· b· b · b.

(ab)3ab3

Let’s expand these exponents and see why this is:

(ab)3ab3

Write out the base ab times itself three times:

(ab)(ab)(ab) ≠ a· b· b· b

The commutative property of multiplication allows us to rearrange the order in which we multiply the a’s and b’s.

a· a· a· b· ba· b· b· b

Rewriting this repeated multiplication we get:

a3b3ab3

The following, though, is true:

(ab3) = ab3

On the right, the a has only an exponent of 1.  If you do not see an exponent written, it is one.  If we write it out we see:

(abbb)=abbb$\left(a\cdot b\cdot b\cdot b\right)=a\cdot b\cdot b\cdot b$

In summary of this first exploration, the base can be tricky to see.  Parenthesis group things together.  An exponent written outside the parenthesis creates all of the terms inside the parenthesis as the base.  But if numbers are multiplying, but not grouped, and one has an exponent, the exponent only belongs to the number just below it on the left.  For example, 4x3,$4{x}^{3},$ the four has an exponent of just one, while the x is being cubed.

Consider: (x+5)3.${\left(x+5\right)}^{3}.$ This means the base is x + 5 and it is multiplied by itself three times.

(x+5)3=(x+5)(x+5)(x+5)(x+5)3x3+53

Repeated Multiplication Allows Us Some Short-Cuts

Consider the expression:

a3×a2.${a}^{3}×{a}^{2}.$

If we wrote this out, we would have:

aaa×aa$a\cdot a\cdot a×a\cdot a$.

(Note:  In math we don’t use colors to differentiate between two things.  A red a and a blue a are the same.  These are colored to help us keep of track of what’s happening with each part of the expression.)

This is three a’s multiplying with another two a’s.  That means there are five a’s multiplying.

a3×a2=a5${a}^{3}×{a}^{2}={a}^{5}$

Before we generalize this to find the short-cut, let us see something similar, but is a potential pitfall.

a3×b2${a}^{3}×{b}^{2}$

If we write this out we get:

aaa×bb$a\cdot a\cdot a×b\cdot b$

This would not be an exponent of 5, in anyway.  An exponent of five means the base is being multiplied by itself five times.  Here we have an a as a base, and three of those multiplying, and a b as a base, and two of those multiplying.  Not five of anything.

The common language is that if the bases are the same we can add the exponents.  This is a hand short-cut, but if you forget where it comes from and why it is true, you’ll undoubtedly confuse it with some of the other short-cuts that follow.

Short-Cut 1:  If the bases are the same you can add the exponents.  This is true because exponents are repeated multiplication and the associative property says that the order in which you group things does not matter (when multiplying).

am×an=am+n${a}^{m}×{a}^{n}={a}^{m+n}$

The second short-cut comes from groups and exponents.

(a3)2${\left({a}^{3}\right)}^{2}$

This means the base is a3, and it is being multiplied by itself.

a3×a3${a}^{3}×{a}^{3}$

Our previous short cut said that if the bases are the same, we can add the exponents because we are just adding how many of the base is being multiplied by itself.

a3×a3=a3+3=a6${a}^{3}×{a}^{3}={a}^{3+3}={a}^{6}$

But this is not much of a short cut.  Let us look at the original expression and the outcome and look for a pattern.

(a3)2=a6${\left({a}^{3}\right)}^{2}={a}^{6}$

Short-Cut 2:  A power raised to another is multiplied.

(am)n=am×n${\left({a}^{m}\right)}^{n}={a}^{m×n}$

Be careful here, though:

a(b3c2)5$a{\left({b}^{3}{c}^{2}\right)}^{5}$ = ab15c10$a{b}^{15}{c}^{10}$

Summary:  Exponents are repeated multiplication.  The superscript (number on the top), indicates how many times the base is multiplying itself.  All of the rules and “laws” of exponents come from this fact.

Practice Problems

 1.    x4⋅x2${x}^{4}\cdot {x}^{2}$ 8.  (5xy)3${\left(5xy\right)}^{3}$ 2.     y9⋅y${y}^{9}\cdot y$ 9.  (8m4)2⋅m3${\left(8{m}^{4}\right)}^{2}\cdot {m}^{3}$ 3.     z2⋅z⋅z3${z}^{2}\cdot z\cdot {z}^{3}$ 10.  (3x5)3(32x7)2${\left(3{x}^{5}\right)}^{3}{\left({3}^{2}{x}^{7}\right)}^{2}$ 4.     (x5)2${\left({x}^{5}\right)}^{2}$ 11.  7(72x4)5⋅73x5$7{\left({7}^{2}{x}^{4}\right)}^{5}\cdot {7}^{3}{x}^{5}$ 5.     (y4)6${\left({y}^{4}\right)}^{6}$ 12. 53+53+53+53+53${5}^{3}+{5}^{3}+{5}^{3}+{5}^{3}+{5}^{3}$ 6.     x3+x3+x3+x8+x8${x}^{3}+{x}^{3}+{x}^{3}+{x}^{8}+{x}^{8}$ 13.  32⋅9${3}^{2}\cdot 9$ 7.     4x+4x+4x${4}^{x}+{4}^{x}+{4}^{x}$ 14. 4x⋅4x⋅4x${4}^{x}\cdot {4}^{x}\cdot {4}^{x}$

Select Practice Problems Review

6.   3x3 + 2x8   The reason this is the answer is because exponents do not change with multiplication.  That is also why number 7 is 3×4x

10.  (3x5)3(32x7)2${\left(3{x}^{5}\right)}^{3}{\left({3}^{2}{x}^{7}\right)}^{2}$ We will work with the exponents outside of the parenthesis first, giving us the following.

(3x5)3(32x7)2=(33x15)(34x14)${\left(3{x}^{5}\right)}^{3}{\left({3}^{2}{x}^{7}\right)}^{2}=\left({3}^{3}{x}^{15}\right)\left({3}^{4}{x}^{14}\right)$

Now we will put the like-bases together by adding their exponents.

(33x15)(34x14)=37x29$\left({3}^{3}{x}^{15}\right)\left({3}^{4}{x}^{14}\right)={3}^{7}{x}^{29}$

12. This one is tricky.  Repeated addition is multiplication, and we have five of the same number being added.

53+53+53+53+53=5×53${5}^{3}+{5}^{3}+{5}^{3}+{5}^{3}+{5}^{3}=5×{5}^{3}$

Now what we see is we have multiplication, giving us 54.

Teaching Exponents

Context: It is very easy to allow students to follow the “rules” of exponents without challenging their thinking.  It is imperative that their conceptual foundation is tested until it is solid because this is one of the first topics taught.  Further, the concept comes heavily into play with exponential and logarithmic functions later.

Big Idea

Exponents are a short-hand way of writing repeated multiplication.  This is often an easier way of communicating the value of a number compared to writing the number out in standard form.

Key Knowledge

There are not laws or rules of exponents, but instead just consequences of the properties of repeated multiplication and division.  If students can revert to the basic facts, the Big Idea, their foundation will be solid and they will not need to worry about forgetting how exponents “work.”

Pro-Tip(for students)

Coach students that when they forget how exponents work to set up simple, easily calculated, hypothesis testers.  For example, if students forget if exponents should be added or multiplied, they can set up a small test to determine what happens and when.  This is a much better tool than brute force memorization.

Lesson

This is a one-day lesson, but the first of many days on the topic of exponents.