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Chapter 3 – Exponents, Radicals and Compound Interest

Chapter 3.1: Exponents and Function NotationIn the previous chapter, we learned that linear growth is given when we have a constant slope, that is, when the same amount is added to the output each time. Which of the input/output tables below show linear growth? If the growth pattern continues in the same way, which company will make the most profit in year 4?

Acme Corp. Bonanza Co. Capital Co. Year Money Year Money Year Money 0 $1000 0 $1 0 $2000 1 $1010 1 $10 1 $2000 2 $1020 2 $100 2 $2000 3 $1030 3 $1000 3 $2000 4 ? 4 ? 4 ?

Each table has a consistent pattern, but not all show a linear growth. Only Acme and Capital show the same amount being added each time, for a constant slope. For Bonanza, the same amount is being multiplied each time, not added.

Acme Corp. Bonanza Co. Capital Co. Year Money Year Money Year Money 0 $1000 0 $1 0 $2000 1 $1010 1 $10 1 $2000 2 $1020 2 $100 2 $2000 3 $1030 3 $1000 3 $2000 4 $1040 4 $10000 4 $2000

Acme shows a change in profit of $10 more dollars per year. This is linear growth, and we can write the linear equation y = 10x + 1000. Notice that the repeated addition of 10 becomes the slope, 10. The slope of 10 multiplies the variable x, because repeated addition of whole numbers is multiplication (for example, 10+10+10+10 = 4 × 10).

In year 4, Acme will have a profit of y = 10x + 1000 =10(4) + 1000 = $1040.

We can also use something called function notation to show Acme’s equation. We can write 𝐴𝐴(𝑥𝑥) = 10𝑥𝑥 + 1000. The first part, A(x), is read “A of x” and just means the output value of the function, A, at the input, x.” We then write A(4)=1040 as a quick way to indicate that when x is 4, the output of the function is 1040.

Capital shows a change in profit of $0 more dollars per year. This is linear growth, because it is a constant change of $0 each time. The slope of this line is zero, and we can write the linear equation y = 2000. We can also write this in function notation as C(x) = 2000. We can name a function using any letter we like – in this case, C for Capital.

In year 4, Capital will still have a profit of y = $2000, ahead of Acme for now, but far behind Bonanza. We can write, C(4) = 2000.

+10+10+10+10

×10 ×10 ×10 ×10

+0+0+0+0

2 Bonanza shows a change in profit of $9 the first year, $90 the next year, and $900 the year after that. This is not linear growth, since the slope changes each time period. The type of growth show in this table is exponential growth. The equation is 𝑦𝑦 = 10𝑥𝑥 , or in function notation, 𝐵𝐵(𝑥𝑥) = 10𝑥𝑥 . Notice that the repeated multiplication of 10 becomes the base, 10, because repeated multiplication of whole numbers is represented by exponents – for example, 10×10×10×10 = 104.

In year 4, we can see that Bonanza will be far ahead of the other two companies, with a profit of 𝑦𝑦 = 10𝑥𝑥 = 104 = 10,000. We can also write B(4)=10,000.

The function notation gives us a quick way to note the results in year 4 for each function: A(4)=1040 C(4) = 2000 B(4)=10,000

Exponential growth can be extremely fast if we are multiplying by a positive number greater than 1. We can see this quick growth in how soon Bonanza has the most profit.

Typically, any type of growth where the new amount is gotten by multiplying the old amount by a constant number is exponential growth. Things that tend to grow exponentially include: population, viruses, the spread of a rumor on the internet, and money in a bank account that gets interest.

The spread of a rumor or a virus:

1 person 3 people 9 people 27 people ….

This graphic shows the number of people tripling each time. You can imagine different rates of growth; for example, an internet meme that goes viral may have one person showing 100 people, then they each show 100 more people, etc. For the equations 𝑦𝑦 = 3𝑥𝑥 and 𝑦𝑦 = 3𝑥𝑥, which is linear? Which is exponential? Which grows more quickly? Before you turn the page, see if you can make a table of input and output values using positive inputs, and then graph each one.

3

The first function, 𝑦𝑦 = 3𝑥𝑥, is an exponential equation. The slope of this graph keeps changing, getting steeper and steeper. Notice that, in order to graph this equation, we needed to change the scale on the y-axis. We make sure we are still going up by an even amount each time, but now we are going up by 5’s on the y-axis instead of by 1’s.

The second function, 𝑦𝑦 = 3𝑥𝑥, is a linear equation. The slope of this graph is the same throughout.

𝑦𝑦 = 3𝑥𝑥 1 31= 3 2 32= 9 3 33= 27 4 34= 81

𝑦𝑦 = 3𝑥𝑥 1 3(1)= 3 2 3(2)= 6 3 3(3)= 9 4 3(4)= 12

Steeper slope up here…

…than here

4 The meaning of zero and negative exponents When we have positive, whole number exponents, they stand for repeated multiplication. For example, 23 means 2 × 2 × 2, which is 8. Similarly, x3= x • x • x.

To understand negative, zero, and non-whole number exponents may require a little more work. The patterns below help extend the positive exponents into zero and negative exponents:

From the table on the left, we can see that, even if we don’t know how to find 24, we know we can just multiply 23 × 2 to get 8 × 2 = 16.

It follows that we can also go backwards up this table, to see what the zero and negative exponents should be.

To get from 24= 16 to 23 , we divide by 2: 16 ÷ 2 = 8, so 23= 8. Similarly, to find 20, we can divide 21 by 2 and we get 2 ÷ 2 = 1.

This means that 20 must equal 1.

This leads us to our first property of exponents:

Property 1 Notice that x cannot equal zero. In fact, 00 is undefined.

𝑦𝑦 = 2𝑥𝑥 -3 -2 -1 0 1 21= 2 2 22= 4 3 23= 8 4 ?

𝑦𝑦 = 2𝑥𝑥 -3 -2 -1 0 ? 1 21= 2 2 22= 4 3 23= 8 4 24= 16

𝑦𝑦 = 2𝑥𝑥 -3 -2 -1 0 1 1 21= 2 2 22= 4 3 23= 8 4 24= 16

×2 ×2 ×2

÷2 ÷2 ÷2 ÷2

𝑥𝑥0 = 1 (𝑥𝑥 ≠ 0)

Any (non-zero) number raised to the zero power is 1.

÷2 ÷2 ÷2 ÷2

5 To get the next property of exponents, we continue the table.

To get from 20= 1 to 2-1, we divide by 2 again: 1 ÷ 2 = ?

On your calculator if you type in 1 ÷ 2 , you get 0.5. But it is easier to see the pattern if we write this result in fraction form: 1÷2 =0.5 = 1

2 , so 2-1= 1

2. Divide by 2 again to get the fractions

and decimals for 2-2 and 2-3 .

Calculator tip: On some calculators, to get 23, you must use the ^ key: .

On others, use the yx or xy key: . Make sure you know which one works with your calculator and not only on your phone, since phones are not allowed on tests and quizzes.

Make sure you also know how to make the exponent negative for 2−3. You will probably have to hit the sign change key after typing the 3.

Exponent

Sign change

𝑦𝑦 = 2𝑥𝑥 -3 ? -2 ? -1 ? 0 20= 1 1 21= 2 2 22= 4 3 23= 8 4 24= 16

𝑦𝑦 = 2𝑥𝑥 -3 2−3 = 0.125 = 1

8

-2 2−2 = 0.25 = 14

-1 2−1 = 0.5 = 12

0 20= 1 1 21= 2 2 22= 4 3 23= 8 4 24= 16

÷2 ÷2 ÷2 ÷2

6 Now you can see a pattern in the ways in which the positive and negative exponents mirror each other.

Notice that 21= 2, while 2−1 = 12

22= 4, while 2−2 = 14

23= 8, while 2−3 = 18

The pattern brings us to the next property of exponents:

Property 2

We can also think of negative exponents as repeated division, while positive exponents are repeated multiplication. The fact that negative exponents are fractions, or reciprocals, also helps us understand the left side of the graph for 𝑦𝑦 = 3𝑥𝑥 that we saw a few pages ago. Notice that the y values on the left side of the graph get smaller and smaller, but the graph never crosses the x-axis. This is because the y-values for 𝒚𝒚 = 𝟑𝟑𝒙𝒙 are never negative numbers, they are fractions!

For example, if we let 𝑥𝑥 = −1 in the equation 𝑦𝑦 =3𝑥𝑥 , we get 𝑦𝑦 = 3−1 = 1

3, which is the point (-1, 1/3)

on the graph.

If we let 𝑥𝑥 = −2 in the equation 𝑦𝑦 = 3𝑥𝑥, we get 𝑦𝑦 =3−2 = 1

32= 1

9, which is the point (-2, 1/9) on the

graph. EXAMPLES – try each one before turning the page! a. (3𝑥𝑥)0 b. (3𝑦𝑦2𝑥𝑥𝑧𝑧3)0

c. 𝑥𝑥−2

d. 3𝑥𝑥−2 e. (3𝑥𝑥)−2 f. 5𝑎𝑎−3𝑏𝑏7

𝑥𝑥−𝑎𝑎 =1𝑥𝑥𝑎𝑎

𝑥𝑥 ≠ 0

A number x, to the exponent, -a, is the reciprocal of the same number raised to a.

�−1,13�

�−2,19�

7 EXAMPLES -- Answers a. (3𝑥𝑥)0 = 1 b. (3𝑦𝑦2𝑥𝑥𝑧𝑧3)0 = 1

Both answers are 1 because anything (except 0) to the zero power is 1, even a whole expression! (𝑎𝑎𝑎𝑎𝑦𝑦𝑎𝑎ℎ𝑖𝑖𝑎𝑎𝑖𝑖)0 = 1

c. 𝑥𝑥−2 = 1

𝑥𝑥2 This example directly uses the property of exponents, above.

d. 3𝑥𝑥−2 = 3

𝑥𝑥2 This example also uses the property of exponents, above. Because the 3

does not have a negative exponent, the 3 stays where it is. e. (3𝑥𝑥)−2 = 1

(3𝑥𝑥)2 = 19𝑥𝑥2

Here, the 3 is also to the -2 power, so both the x and the 3 are “flipped.” g. 5𝑎𝑎−3𝑏𝑏7 = 5𝑏𝑏7

𝑎𝑎3

The only part of this that gets “flipped” is the part with the negative exponent. Notice that after you take the reciprocal, the exponent becomes positive.

h. 5

𝑥𝑥−4= 5𝑥𝑥4

1= 5𝑥𝑥4

A negative exponent that is in the denominator can also be “flipped,” but now it goes up to the numerator and becomes positive.

Some people are very surprised by this one! They say, “But I thought the 3 was attached to the x!” But it’s not, really, it’s

just three multiplied by 𝑥𝑥−2.

When you have a negative exponent, cross the line and change the sign!

8 There are three addition properties of exponents. Using the examples below, let’s see if you can figure them out before you turn the page! Property 3: What happens to the exponents when we multiply numbers that have the same base? For example, what is 23 ∙ 25?

You may have some guesses as to what you will get. Maybe the answer is 28? Or maybe, it’s 215? To figure out which answer is correct, consider how many 2’s you have when you expand each of these: 23 = 2 ∙ 2 ∙ 2, and 25 = 2 ∙ 2 ∙ 2 ∙ 2 ∙ 2, so we have 23 ∙ 25 = (2 ∙ 2 ∙ 2) ∙ ( 2 ∙ 2 ∙ 2 ∙ 2 ∙ 2), which is 28 in total.

See if you can now write this pattern as a general property in words and using variables: When we multiply numbers or variables of the same base, we __?_ the exponents. 𝑎𝑎𝑥𝑥 ∙ 𝑎𝑎𝑦𝑦 =? Property 4: The property for division of numbers with exponents is, in some ways, the opposite of the property for multiplication. To see what to do, we again turn to an example.

Let’s simplify 25

23 by expanding out the numerator and denominator, and seeing which

numbers cancel. 25

23=

2 ∙ 2 ∙ 2 ∙ 2 ∙ 22 ∙ 2 ∙ 2

= 2 ∙ 2 = 22

See if you can now write this pattern as a general property in words and using variables: When we divide numbers or variables of the same base, we __?___ the exponents. 𝑎𝑎

𝑥𝑥

𝑎𝑎𝑦𝑦= ?

Property 5: What happens to the exponents when we raise an exponent to another exponent, for example, (25)2?

To see what to do, remember that when we raise an expression to the second power this means that we are repeating it twice. That is, (25)2 = 25 ∙ 25. We can use our previous property, or we can expand this out, to get that 25 ∙ 25 = 210.

Thus, going back to the original question, we now know that (25)2 = 210. What happened to the original exponents, and how would you write this as a general property?

9 Property 3

Property 4

Property 5

EXAMPLES: Simplify the following completely, using the correct property, above.

a. 𝑚𝑚4 ∙ 𝑚𝑚3 ∙ 𝑚𝑚 = 𝑚𝑚4+3+1 = 𝑚𝑚8 Note that if a base has no exponent, the exponent is assumed to be a 1, so that 𝑚𝑚 = 𝑚𝑚1

b. (4𝑎𝑎5)(3𝑎𝑎11) = (4)(3)(𝑎𝑎5+11) = 12𝑎𝑎16 The coefficients should be multiplied,

because this is the operation we are performing, and exponents are added.

c. 15𝑚𝑚6

3𝑚𝑚3 = (153

)𝑚𝑚6−3 = 5𝑚𝑚3 The coefficients should be divided, because this is the operation we are performing, and exponents are subtracted.

d. 3𝑚𝑚3

15𝑚𝑚6 = � 315�𝑚𝑚3−6 = 1

5𝑚𝑚−3 = 1

5𝑚𝑚3 This one is a little trickier. To keep our

results as whole numbers, and not decimals, we simplify 3

15 by dividing

both numerator and denominator by 3. Next, we subtract exponents. We end up with a negative exponent 𝑚𝑚−3, so we take the reciprocal and make the exponent positive.

𝑎𝑎𝑥𝑥 ∙ 𝑎𝑎𝑦𝑦 = 𝑎𝑎𝑥𝑥+𝑦𝑦 𝑎𝑎≠ 0

This property tells us when the base is the same, and we are multiplying exponential expressions, we can combine exponents by adding them.

𝑎𝑎𝑥𝑥

𝑎𝑎𝑦𝑦= 𝑎𝑎𝑥𝑥−𝑦𝑦 𝑎𝑎≠ 0

This property tells us when the base is the same, and we are dividing exponential expressions, we can combine exponents by subtracting them.

(𝑎𝑎𝑥𝑥)𝑦𝑦 = 𝑎𝑎𝑥𝑥𝑦𝑦 𝑎𝑎≠ 0

This property tells us when the base is the same, and we are multiplying exponential expressions, we can combine exponents by multiplying them.

10 Another way to do this is to realize there are more m’s in the denominator. We can imagine the m’s “canceling”: so that the remaining m’s will be in the denominator:

3𝑚𝑚3

15𝑚𝑚6 = �3𝑚𝑚𝑚𝑚𝑚𝑚

15𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚� =

15𝑚𝑚𝑚𝑚𝑚𝑚

=1

5𝑚𝑚3

e. (𝑎𝑎5)4 = (𝑎𝑎5∙4) = 𝑎𝑎20 Multiply exponents, as in property 5. f. (3𝑏𝑏5)2 = (32)(𝑎𝑎5∙2) = 9𝑎𝑎10 The 3 is just to the second power,

while the exponents are multiplied. Notice that in this last example, both the number and variable got the second power. We can think of the variable as being “distributed” to each item inside the parenthesis. This happens because the power on the outside is telling us how many times to distribute everything on the inside: (3𝑏𝑏5)2 = 3𝑏𝑏5 ∙ 3𝑏𝑏5 = 9𝑎𝑎10.

In the next three examples, we will again see how parenthesis means the exponent is distributed throughout:

g. �𝑥𝑥4�3

= (𝑥𝑥4)(𝑥𝑥4)(𝑥𝑥4) = 𝑥𝑥3

43= 𝑥𝑥3

64 We can skip the step where we repeat 𝑥𝑥

4,

and instead just think of the 3rd power as being distributed to everything inside the ( ).

h. �𝑥𝑥5𝑥𝑥2

𝑥𝑥3�4

= 𝑥𝑥20𝑥𝑥8

𝑥𝑥12= 𝑥𝑥28

𝑥𝑥12= 𝑥𝑥16

OR here is another way to simplify this expression:

�𝑥𝑥5𝑥𝑥2

𝑥𝑥3�4

= �𝑥𝑥7

𝑥𝑥3�4

= (𝑥𝑥4)4 = 𝑥𝑥16 We can first combine numerators by adding exponents (property 3), then combine the exponents in the quotient by subtracting them (property 2), then raising a power to a power means we multiply exponents (property 5).

There are often many correct ways to simplify an

exponential expression. Just make sure you use the

properties properly!

11 EXAMPLES: Let’s put all of the properties together, and simplify each of the following. Express all answers using only positive exponents. Try working out each example on your own, before looking at the answer, below. a. (2𝑥𝑥−3𝑦𝑦5𝑧𝑧−2)(−3𝑥𝑥6𝑦𝑦−8𝑧𝑧2)

b. (𝑥𝑥−8)�𝑥𝑥15�

(𝑥𝑥10)3

c. �4𝑥𝑥−4𝑦𝑦5

10𝑥𝑥𝑦𝑦−2�3

d. �3𝑎𝑎5

𝑎𝑎−4�−2

Answers:

a. (2𝑥𝑥−3𝑦𝑦5𝑧𝑧−2)(−3𝑥𝑥6𝑦𝑦−8𝑧𝑧2) =

(2)(−3)(𝑥𝑥−3+6)(𝑦𝑦5−8)(𝑧𝑧−2+2)=

−6𝑥𝑥3𝑦𝑦−3𝑧𝑧0=−6𝑥𝑥3𝑦𝑦−3 ∙ 1

−6𝑥𝑥3𝑦𝑦−3 = −6𝑥𝑥3

𝑦𝑦3

b. (𝑥𝑥−8)�𝑥𝑥15�

(𝑥𝑥10)3= 𝑥𝑥7

𝑥𝑥30

= 𝑥𝑥7−30 = 𝑥𝑥−23

=1𝑥𝑥23

c. �4𝑥𝑥−4𝑦𝑦5

10𝑥𝑥𝑦𝑦−2�3

= 43𝑥𝑥(−4)(3)𝑦𝑦(5)(3)

103𝑥𝑥3𝑦𝑦(−2)(3)

= 64𝑥𝑥−12𝑦𝑦15

1000𝑥𝑥3𝑦𝑦−6

= 64𝑥𝑥−15𝑦𝑦21

1000

=8𝑦𝑦21

𝑥𝑥15

Multiply coefficients, and add exponents. 𝑧𝑧0 = 1 The exponent on the y is negative 3. To make it positive, “flip” it into the denominator.

Add exponents in the numerator: -8 + 15 = 7; multiply exponents in the denominator: 10 x 3 = 30. Next, subtract exponents: 7-30 =-23 Finally, express the negative exponent as a positive by taking the reciprocal.

Raise each number or variable inside parenthesis to the third power; multiply the exponents.

Now we want to combine numbers, x variables, and y variables.

We combine exponents on the x’s by subtracting: −12 – 3 = −15. We also combine exponents on the y’s by subtracting: 15 − (−6) = 15 + 6 = 21.

We can simplify 64 and 1000 by dividing both by 8.

To make the negative exponent on 𝑥𝑥−15 positive, bring it to the denominator.

12

d. �3𝑎𝑎5

𝑎𝑎−4�−2

There are several ways to do this one! One possible method is to flip the whole fraction. Once you flip it, the exponent becomes postive:

�3𝑎𝑎5

𝑎𝑎−4�−2

= �𝑎𝑎−4

3𝑎𝑎5�2

Remember, once you take the reciprocal, the exponent becomes positive – remember our pattern from page 5! 𝑥𝑥−2 = 1

𝑥𝑥2

�𝑎𝑎−4

3𝑎𝑎5�2

=𝑎𝑎−8

9𝑎𝑎10

𝑎𝑎−8

9𝑎𝑎10 =𝑎𝑎−8−10

9 =𝑎𝑎−18

9

=1

9𝑎𝑎18 Alternatively, we could start by distributing the exponent of -2 to to each part inside the ( ):

�3𝑎𝑎5

𝑎𝑎−4�−2

=3−2(𝑎𝑎5)−2

(𝑎𝑎−4)−2 =3−2𝑎𝑎−10

𝑎𝑎8

=1

32𝑎𝑎8𝑎𝑎10

=1

9𝑎𝑎18

Now multiply exponents: −4 ∙ 2 = −8 and 5 ∙ 2 = 10

and raise 3 to the 2nd power: 32 = 9 Subtract exponents. Finally, write in terms of postive exponents.

CAUTION!!! 3−2 is not -9!

3−2 =19

Now multiply exponents: 5 ∙ −2 = −10 and −4 ∙ −2 = 8

and raise 3 to the -2nd power. Bring any negative exponents down and make them positive. Finally, combine exponents in the denominator by adding: 𝑎𝑎8𝑎𝑎10 = 𝑎𝑎18