UNDERSTANDING THE RULES FOR EXPONENTS

Exponents play a large role in algebra. If you are not fully comfortable with what an exponent is, and the rules for exponents of 0 and 1, and how to do basic computa- tions with exponents, review the corresponding material in the first book before proceeding with this chapter. This chapter presents some rules for how to handle exponents in algebraic expressions and equations. It is very easy to confuse one rule with another. Instead of trying to memorize the rules, it is far better to under- stand how they work. In doing so, you will be able to recreate a rule anytime that you need to.

PROBLEMS OF THE FORM

𝒙

∙ 𝒙

The first rule tells us how to multiply two exponential terms which have the same base, but different exponents, for example, 𝑥2∙ 𝑥3. The rule is that in situations like this, we add the exponents and keep the base. The

example simplifies to 𝑥5. As mentioned, though, rather than memorize the rule, it is far better to understand it. Think about what 𝑥2 means. It means x · x. We also know that 𝑥3 means x · x · x. What happens when we multiply those two results? We have x · x · x · x · x, which is 𝑥5. In this expression, we really just needed to count how many copies of x would end up being multiplied together.

𝑥

_{∙ 𝑥}

_{= 𝑥}

When we multiply two like bases, we add the exponents

Let’s look at a simple example using actual numbers. Simplify: 124∙ 125. Following the rule, the answer would be 129, which is how you would leave your answer.

“TRICK” PROBLEMS OF THE FORM

𝒙

+ 𝒙

It is important to understand that the previous section involved multiplying and not adding exponential terms with like bases. There is no formula for problems of the form 𝑥𝑎 + 𝑥𝑏 in which we are adding like bases.

Let’s look at an example: Simplify 24+ 28. If we actual- ly compute the values of each term we get 16 + 256 which equals 272. There is no simple formula involving the exponents which would get us to that value. In particu- lar, the answer is not equal to 212 which equals 4096.

E X P O N E N T S I N A L G E B R A

Don’t get tricked by problems of this form. If you are expected to get a numerical answer, for now you will need to just compute it like we did in the example.

PROBLEMS OF THE FORM

𝒙

÷ 𝒙

The next rule we’ll look at is how to divide two exponen- tial terms which have the same base, but different exponents. An example is 𝑥5÷ 𝑥3. The rule is that in situations like this, we subtract the exponents and keep the base. The example simplifies to 𝑥2. As mentioned, though, rather than memorize the rule, it is far better to understand it.

Think about what 𝑥5 means. It means x · x · x · x · x. We also know that 𝑥3 means x · x · x. What happens when we set up the problem as a fraction, and divide the first result by the second, and “cancel” pairs of x’s which appear both the numerator and denominator?

“Cancelling” pairs of common factors to see what remains

Remember that when we “cancel” common factors in a fraction, those factors don’t just disappear. Anything of the form x/x can be reduced to 1/1, which is further

simplified to 1, and in multiplication, 1 is the identity element or “invisible” number which we need not write. In this example, all three x’s in the denominator are paired up with three of the x’s in the numerator. In the denominator we are left with our implied 1 which we can ignore, and in the numerator we are left with x · x. The problem of 𝑥5÷ 𝑥3 simplifies to 𝑥2.

In this expression, we really just needed to count how many copies of x would remain after matching pairs in the numerator and denominator were removed. Notice that what we really did was subtract in order to find the difference between the number of x’s in the numerator and the number of x’s in the denominator.

𝑥

÷ 𝑥

= 𝑥

When we divide two like bases, we subtract the exponents

PROBLEMS OF THE FORM

(𝒙

)

The next rule we’ll look at is how to handle an exponen- tial term which itself is raised to a power. An example is (𝑥2₎3_{. The rule is that in situations like this, we multiply}

the exponents and keep the base. The expression above simplifies to 𝑥6. As mentioned, though, rather than memorize the rule, it is far better to understand it.

E X P O N E N T S I N A L G E B R A

Remember that 𝑥2 means x · x. We also know that when we raise a quantity to the power of 3, it means to multip- ly it by itself so that we have three copies. What happens when we take (x · x) and multiply it times itself for a total of three copies? We have (x · x) · (x · x) · (x · x). We can remove the parentheses which were only added for clarity, and what is left is just x multiplied by itself for a total of 6 copies which is 𝑥6. In this expression, we really just needed to count how many copies of x would end up being multiplied together.

(𝑥

)

= 𝑥

When we raise an exponent term to a given power, we multiply the exponents

Remember: In all problems involving exponents, a

variable by itself such as x should be treated as 𝑥1.

PROBLEMS OF THE FORM

𝒙

The next rule we’ll look at it how to handle a variable raised to a negative exponent. The rule is that we make the exponent positive, and then put that value under a numerator of 1. While it is easy to remember that rule, it is beneficial to have a sense of where it comes from. Let’s look at the problem 𝑥3÷ 𝑥5. We can set it up as shown below. Notice how we can “cancel” pairs of x’s

like we did in an earlier problem. On top we are just left with an implied 1, and on the bottom we are left with 𝑥2. This shows that 𝑥3÷ 𝑥5 = 1/𝑥2.

“Cancelling” pairs of common factors to see what remains

With that in mind, remember that we could have solved this problem using the formula which tells us to subtract the exponents since we are dividing like bases. If we did that (in the proper order), we would have 𝑥3−5 or 𝑥−2. Informally, this shows that 𝑥−2 is the same as 1/𝑥2, and of course the pattern holds for any negative exponent.

𝒙

=

𝒙𝒂

When a value is raised to a negative exponent, make the exponent positive, and put the value under a numerator of 1

SO WHY DOES

𝒙

In the first book we said that any value raised to the power of 0 equals 1, but no explanation was offered. Now that we are learning about how exponents work in algebra, an informal proof by example can be provided. Let’s look at the problem 𝑥3÷ 𝑥3. We’re dealing with the situation of a quantity divided by itself which we know

E X P O N E N T S I N A L G E B R A

is always equal to 1. With that in mind, remember that we could have solved this problem using the formula which tells us to subtract the exponents since we are dividing like bases. If we did that we would have 𝑥3−3 or 𝑥0_{. Informally, this shows that 𝑥}0 _{is equal to any problem}

of the form 𝑥𝑎÷ 𝑥𝑎 which in turn is equal to 1.

MULTIPLYING TERMS WITH COEFFICIENTS AND