4. Exponent Operations: Podcasts that spread like the plague (that’s a good thing...)

Chapter 4. Exponent Operations: Podcasts that spread like the plague (that’s a good thing...)

Could you multiply that again? Could you multiply that again?

There’s another way to express multiplication that’s repeated over and over and over again, without just repeating yourself. Exponents are a way of repeating multiplication. But there’s more to exponents, including some smaller-than-usual numbers (and we don’t just mean fractions). In this chapter, you’ll brush up on bases, roots, and radicals, all without getting arrested for any sit-in protests. And, as usual, zero and one come with their own problems... so jump into a podcasting exponentiation extravaganza.

Addie’s got a podcast

Note

Ahem, eccentric

Addie podcasts about crazy celebrities.

Addie’s been getting a lot more listeners lately. To take it to the next level, she needs new equipment to deliver an even better podcast... but that takes a lot of cash.

Addie’s got a website to host her podcasts, and she wants to advertise on her site to raise funds for new gear. She’s lined up some possible sponsors, but they won’t help out until Addie proves she can get some real traffic on her site. Addie needs to:

...monitor the daily hits on her website over the next 2 weeks.
...prove her site can generate at least 5,000,000 hits in the next 2 weeks!
Note
Wow, that’s a lot.

Let’s mobilize Addie’s listeners

Addie knows she’s got big fans. Here’s a letter she’s worked up to send out to her 3 top listeners:

Exponents are a special notation used to express repeated multiplication. That’s just what we need to figure out Addie’s number of hits without counting a lot of threes: a way to show multiplying by 3 over and over.

When you see a number with an exponent, it looks like this:

The base is the number being multiplied (in Addie’s case, 3), and the exponent is the number of times you repeat it (in Addie’s case, 14). Those two numbers are all you need to put in your calculator, and you’ll get the answer.

Can Addie and Alex get enough hits?

Alex has offered to send off another round of emails for Addie. He’ll start with 3 friends, just like she did, and try to help get 5,000,000 hits in 14 days.

To figure out the total number of hits, you’ll need to figure out how to add up both groups that Addie’s working with. In Chapter 2, you combined like terms to help Paul on his road trip, and this is the same idea. You may remember from Chapter 2 that a term is any part of an equation held together with multiplication or division. Since an exponent is just a shorthand version of multiplication, that means exponential terms with the same base and the same exponent are like terms.

With exponents, you can combine terms that have the same base. Let’s try that out and see how it works:

Because (3¹⁴)² is multiplication, not addition.

A term is something held together by multiplication, which means that the entire exponential term is treated as a group.

Note

3¹⁴ is one term, not two.

When you group two like terms together, you’re adding those terms together. But if you take those two terms and use exponents (that ² at the end of (3¹⁴)²), then you’re multiplying, and that’s not what we want. Look:

Well, that would work, but that’s a lot of threes...

Writing out the multiplication by hand will work; it’s just not very convenient. Look how long this thing turns out to be:

But look, there’s a pattern! Here’s what this means:

Exponents with the same base are LIKE TERMS.

That means they can be added, subtracted, multiplied, and divided.

Try division:

Alex is flaking out on his sister

Alex didn’t send out any emails to his friends until the third day. That means he only has 12 days to get the word out. Will Addie still make it?

Q:
Q: : Why use exponents and not just multiplication?
A:
A: Because it can save you a bunch of work. Writing out a value to multiply over and over again is tedious and leads to error. And when the numbers start to get really big (like an exponent of 14), they are just impossible to deal with otherwise.
Q:
Q: Why do you need the same base and the same exponent to do addition and subtraction?
A:
A: Because they need to be like terms. Remember that exponents are a shorthand for multiplication. Because of the order of operations, you can’t add two multiplication expressions together without doing the multiplications first... unless you’ve got like terms. If the expressions are like terms, then you can collect them together into a single term. That’s exactly what adding exponential terms with the same base and exponent does!
Q:
Q: Where are exponents in the order of operations?
A:
A: They’re second. Since exponents are just a more powerful form of multiplication, they go before multiplication. So it’s parentheses, exponents, then multiplication and division.
Q:
Q: How do you work with exponents with different bases?
A:
A: We’re going to be looking at those next. But fair warning, there’s not much you can do to make those problems simpler. If you have two bases, then there are two things that need to be multiplied, divided, or whatever. There’s not a good way to combine terms like that since you’ve got to keep track of both bases separately.
Q:
Q: What happens if I’m dividing exponential terms and the exponent becomes negative?
A:
A: Great question! When you divide exponential terms, you subtract the exponents. This means that you could end up with a negative exponent. The good news is that this is easy to deal with. A negative exponent just means 1 over the positive exponential term. So:
2^-1 is 1/2,
x^-25 is 1/x²⁵
...and so on.

There’s always a villain...

The Movie Podcast heard about Addie’s plan to increase subsribers, and they don’t like it. The sponsorship Addie’s trying to get... well, it’s money out of the Movie Podcast’s pocket. So now they’re fighting back.

Every person that hits Movie Podcast’s page instead of Startalk Podcast’s page is taking away potential hits. What does this mean for Addie’s chances to score a new sponsorship deal?

Different bases = NOT like terms.

Terms with different bases are not like terms (regardless of the exponent). They just don’t have anything in common. As exponential terms, they’re not multiplying the same number, regardless of how many times.

Note

As we saw earlier, they’re only like terms if the base AND the exponent are the same.

You can’t add exponents with different bases

If we just talk about the bit of Addie’s equation that has two terms, it looks like this:

You know that you can’t add or subtract these two because they’re not like terms. Multiplication and division don’t work either. Exponential terms being multiplied together just get written together, like this:

The order of operations says exponents FIRST

You can’t split up bases and combine different exponents because each base has to stay with its own exponent. The order of operations says that exponents go before multiplication. That means the exponents have to be simplified before they can be combined with something else.

Brain Power

Test it out with real numbers - try 3² and 4³. Can you show that (3²)(4³) is not the same as ((3)(4))⁽²⁾⁽³⁾ without working things all the way out to the answer?

Q:
Q: Do I really have to memorize all of these rules for working with exponents?
A:
A: No, because you can always work through these equations by working out each term separately. But if you can remember these rules, you’ll be able to combine like terms and solve equations more quickly. It’s much easier to combine terms and do one calculation. That’s a lot better than working out a ton of terms separately, especially if the terms can be combined because they’re like terms.
Q:
Q: What if the bases are different and the exponents are the same?
A:
A: Well, there’s a little bit you can do there. If the exponents are the same, then each term is being multiplied the same number of times, so you CAN mush them together, like this: x^a • y^a = (xy)^a It only works because of the commutative and associative properties. This is all just multiplication, so you can mix up the order, and it will still work.
Q:
Q: So what about that Brain Power? How could you show it without solving the math?
A:
A: You can do it with variables, so instead of 3 and 4, let’s use x and y. Then we have x²y³ = x•x•y•y•y and (xy)⁽²⁾⁽³⁾=(xy)⁶=xy•xy•xy•xy•xy•xy = x•x•x•x•x•x•y•y•y•y•y•y You can just look at those two and see they’re not equal.

Addie needs another round of emails.

But how many does she need to send out? Addie has only 9 days left and she needs to figure out how many emails she needs to start with today to make up for the campaign that Movie Podcast’s running.

We’ve got to work our exponent “backward”

Let’s go back to Addie’s equation. We’ve got different information this time: the number of hits we need and the number of days left:

Now we fill in the things we know:

A root is the INVERSE of an exponent

We need an operation that can unravel an exponent. So when we have the exponent, what’s the base to get a certain answer? Well, that’s the root. When you find a root of something, you’re finding the number that can be multiplied over and over to reach the final number.

For Addie, we need take the ninth root of both sides of her equation. That will isolate x and give us a numeric value on the other side.

It’s time to pull out a calculator. Look closely—you can punch any root of any number in there and get a solution. Most calculators have a way to punch in both a root key and the root you want (like 9 or 3).

Note

Ask a teacher, or look this up in your calculator’s instructions.

Let’s look at a little closer at roots:

If you wanted to read this, you’d say, “The second root of nine is three.” The actual root is three, and it’s the number that you multiply twice by itself to find the number under the radical. So, to find the ninth root for Addie, all you need is a calculator.

It’s the moment of truth. How many more emails does Addie need to send out? Is there any way she can pull this off?

Solve Addie’s root: _____________________________________________________

_____________________________________________________________________

How many emails does Addie need to send out? ________________________________

______________________________________________________________________

Is there any way she can pull it off?

Yes

Why? _____________________________________________________________________

9 days later...

You’ve helped Addie land a big check!

Addie’s site cleared 5,000,000 hits, no problem. Her sponsorship deals on, the subscribers are pouring in, and Addie’s off to get some great new gear from her local Apple store. Next up... a video campaign on YouTube!

Q:
Q: Just put the problem in a calculator? Is that for real?
A:
A: There are actually several ways to find roots of numbers. There are tables where you can look them up, there’s even a way to find them by hand that looks like long division. But honestly, they’re all really old school. For most folks, a calculator is perfect.
Another way to get near the root of a number is to remember the perfect squares (2x2=4, 3x3=9, etc.). Then you can get an idea of what numbres might at least be close to what you’re looking for.
Q:
Q: What’s the inverse operation of exponents? The radical?
A:
A: Not quite. It’s finding the root. The radical is the symbol for the operation. It’s just like the dot symbolizing multiplication.
Q:
Q: What if I see a radical without an index number?
A:
A: Assume an index of 2. That’s the square root. It’s convention that if there isn’t an index, then the equation is talking about the square root.
Q:
Q: Can you have a fractional exponent?
A:
A: Yes. That simply means you should take the root of the base. For example, if you see 1/2 as an exponent, it means square root. 1/3 would be the third root, and so on.
Q:
Q: My calculator doesn’t have a 9th root button, what do I do?
A:
A: You can write a root as fractional exponent. So, a ninth root can be written as or 734,166^(1/9)
Most calculators have an exponent button. So you could just put in a root of (1/9) and get the same answer.
Q:
Q: Will I ever need to solve for an exponent and not the base?
A:
A: Not anytime soon. There are more operations out there that you can use to do this sort of problem, but they’re well beyond this book. Don’t worry about it for now. (Isn’t that good to hear!)
Q:
Q: What about an exponent of 0?
A:
A: Any number raised to the 0 power is one. Why? If you go back to the division of exponents, you subtract the bottom exponent from the top exponent. If you end up with the same term on the top and the bottom, then it’s the base to the 0 power. That’s always the number “1.”
Q:
Q: What about an exponent of 1?
A:
A: Any number raised to an exponent of 1 is itself. That means an exponent of one is implied over EVERY number and EVERY variable. It can come in handy sometimes to know that.
Q:
Q: Can an exponent be negative?
A:
A: Yes - it means that it’s the exponent in the denominator. So
That ties right in with subtracting exponents again. Since there’s no exponent in the numerator, it’s a negative exponent.
Q:
Q: Can you use negative exponents to get rid of fractions?
A:
A: Yes. If you have an expression with fractions in it, you just rewrite the expression with the denominators as negative exponents. This really only helps if you find working with exponents easier than working with fractions. Of course, some people prefer that, and it’s a perfectly okay way to work.
This also works the other way: if you think fractions are easier than exponents, just pull out all your negative exponents and rewrite them as fractions.
Q:
Q: I’ve heard of something called the principle root. What’s that about?
A:
A: When we talk about finding roots, we’re actually talking about finding the principle root. That’s the positive root of a value. There are other roots to numbers, too, though. The most common is the negative root. For example, the principle square root of 9 is 3, but -3 is a square root of 9, too, since (-3)(-3) = 9.

Raise yourself! Can you get all the words? They’re all from this chapter.

Across		Down
3.	Another word used for exponent is	1.	Exponents with the same base and exponent are
5.	The inverse operation of an exponent is a	2.	A negative exponent means that the exponential term can be written as a
6.	Exponents are a faster form of	4.	The number of times that the base gets multiplied is the
8.	Any number raised to an exponent of zero is	5.	A fractional exponent is actually a
9.	The number in an exponential term that gets multiplied	7.	Any number raised to an exponent of one is

Tools for your Algebra Toolbox

This chapter was about numeric properties that are important to understand in order to work with equations.

Exponents are shorthand for repetitive multiplication.
The base is the number that gets multiplied.
The exponent is how many times the base is multiplied.
To add or subtract terms with exponents, they must have the same base and the same exponent.
Adding and subtracting those terms is just combining like terms.
To multiply exponential terms with the same base, just add the exponents.
To divide exponential terms with the same base, subtract the exponents.
To raise an exponential term to an exponent, multiply the exponents.
Rules for dealing with exponents apply to numbers and variables.

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Head First Algebra by Dan Pilone, Tracey Pilone

Chapter 4. Exponent Operations: Podcasts that spread like the plague (that’s a good thing...)

Addie’s got a podcast

Note

Note

Let’s mobilize Addie’s listeners

Can Addie and Alex get enough hits?

Note

Alex is flaking out on his sister

There’s always a villain...

Note

You can’t add exponents with different bases

The order of operations says exponents FIRST

Brain Power

We’ve got to work our exponent “backward”

A root is the INVERSE of an exponent

Note

Tools for your Algebra Toolbox

Don’t leave empty-handed

It’s yours, free.

Check it out now on O’Reilly