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Lesson 103: More About Exponents
You've already learned the basics of exponents. This lesson will review some facts, and teach you some more. Make sure that you study this important topic.
You've already learned that 25 means to multiply 2
times itself 5 times. This means
2 · 2 · 2 · 2 · 2, which is
32. It is read as "two to the power of 5", or "two to the
fifth power." Now we'll learn how to raise negative numbers to
various powers.
What is (-2)2? That means -2 · -2. We've learned that negative times negative equals positive, so the answer is 4. What is (-2)3? That means -2 · -2 · -2. That is 4 · -2. We've learned that negative times positive equals negative. The answer is -8.
This means that if you raise a negative number to a power which is even, the answer will be positive. If you raise a negative number to a power which is odd, the answer will be negative. Memorize that fact.
Very often we need to multiply numbers with exponents. There is a rule that we can apply when the numbers being multiplied have the same base, meaning the number that is being raised to a power. For example, what is 43 · 45? We know that 43 means 4 · 4 · 4. We know that 45 means 4 · 4 · 4 · 4 · 4. We're multiplying both of those results together. That means we are really multiplying 4 eight times. How can we show that using an exponent? We can write 48, and of course we could evaluate it if we needed to.
If you look carefully, you can see that we actually added
the exponents, and kept the base. What this means is that when
we multiply two numbers with the same base, we add the exponents.
You can see this symbolically at left.
You might be able to guess the rule for division. Let's say we have 75 / 72. What we really have is (7 · 7 · 7 · 7 · 7) / (7 · 7). Two of the sevens in the numerator cancel out with the two sevens in the denominator. What is left? 7 · 7 · 7, or 73.
What we did was subtract
the exponents, and kept the base. What this means is that when
we divide two numbers with the same base, we subtract the exponents.
You can see this using symbols at left.
A
product raised to a power is equal to the product of each factor
with that exponent applied to it. In symbols, we can say that
(ab)x = axbx. This makes
sense because either way, we are multiplying x occurrences of a
times x occurrences of b.
A
quotient raised to a power is equal to the quotient of the dividend
and the divisor with that exponent applied to each. In
symbols, we can say that (a / b)x = ax / bx.
This makes sense because either way, there are x occurrences of a in
the numerator, and x occurrences of b in the denominator.
Any number to the power of 0 equal 1.
Stated another way, x0
= 1. For
example, 50 = 1. The proof is omitted, but see if you can figure out why this
is the case. There are two different ways of showing that this
rule makes sense.
We sometimes have to raise a variable with an exponent to a given
power. An example would be (x2)3.
Let's look at what this means. It means x2 · x2
· x2. Remember that to do this, we will add the
exponents. What we really have is x, multiplied by itself x
times. That's x6. Looking back to our
original (x2)3, you can see that what we
really did is multiply the exponents. Here is our
rule for raising a term to a given power: (xa)b
= xab. This is
tricky. Make sure you see how this is different than the
operations that we did above.
The final rule for exponents is for raising a base to a negative
power.
The rule is: x-a
= 1/(xa). In words, a
base raised to a negative power is equal to 1 divided by the base to
the positive of that power. The proof is omitted, but try to
figure out on your own why this is the case. There are two
different ways of showing that the rule makes sense.
You'll have much more practice later working with exponents. Be sure to memorize all the rules presented in this lesson.
Remember that you can ask a math question if you have additional questions about a topic, or you can contact me if you have any comments or suggestions for this site.