Number Unit

Part 1

Reading Math

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

2.      Multiplication is repeated addition

3.      Addition is “skip” counting

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

e still know things about this expression. 

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 5^th 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:

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

$\left(a\cdot a\right)\left[\left(a\cdot a\cdot a\right)\left(b\cdot b\cdot b\right)\right]$

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

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

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)³ ≠ ab³

 

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

(ab)³ ≠ ab³

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· b ≠ a· b· b· b

Rewriting this repeated multiplication we get:

a³b³ ≠ ab³

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:

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

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, $4x^3,$ the four has an exponent of just one, while the x is being cubed.

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

$\begin{array}{l}{\left(x+5\right)}^3=\left(x+5\right)\left(x+5\right)\left(x+5\right)\\ {\left(x+5\right)}^3\ne x^3+5^3\end{array}$

Repeated Multiplication Allows Us Some Short-Cuts

Consider the expression:

$a^3\times a^2.$

If we wrote this out, we would have:

$a\cdot a\cdot a\times a\cdot a$.

 (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.

$a^3\times a^2=a^5$

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

$a^3\times b^2$

If we write this out we get:

$a\cdot a\cdot a\times b\cdot b$

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).

$a^m\times a^n=a^{m+n}$ 

The second short-cut comes from groups and exponents. 

${\left(a^3\right)}^2$ 

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

$a^3\times a^3$ 

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.

$a^3\times a^3=a^{3+3}=a^6$

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

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

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

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

Be careful here, though:

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

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.    $x^4\cdot x^2$	8.  ${\left(5xy\right)}^3$
2.     $y^9\cdot y$	9.  ${\left(8m^4\right)}^2\cdot m^3$
3.     $z^2\cdot z\cdot z^3$	10.  ${\left(3x^5\right)}^3{\left(3^2{x}^7\right)}^2$
4.     ${\left(x^5\right)}^2$	11.  $7{\left(7^2{x}^4\right)}^5\cdot 7^3{x}^5$
5.     ${\left(y^4\right)}^6$	12. $5^3+5^3+5^3+5^3+5^3$
6.     $x^3+x^3+x^3+x^8+x^8$	13.  $3^2\cdot 9$
7.     $4^x+4^x+4^x$	14. $4^x\cdot 4^x\cdot 4^x$

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 3×4^x. 

 

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

${\left(3x^5\right)}^3{\left(3^2{x}^7\right)}^2=\left(3^3{x}^{15}\right)\left(3^4{x}^{14}\right)$

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

$\left(3^3{x}^{15}\right)\left(3^4{x}^{14}\right)=3^7{x}^{29}$

 

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

$5^3+5^3+5^3+5^3+5^3=5\times 5^3$

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