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## Multiplying and Dividing Monomials
The main ideas in this lesson are the laws for multiplying and dividing powers. In this lesson, we will deal with monomials that are powers of a single variable. ## PowersLet's begin by reviewing powers and exponents. If a is a
variable, then a a a . . . a . b factors Remember that a is called the ## Laws of Exponents and Multiplying MonomialsWhen multiplying monomials, we must analyze the product of the
two powers of the same base. Consider x 2 2 In both cases, the exponent of the resulting power is the sum
of the exponents in the two factors. For 2 Notice each power that results. Do you see a pattern? Each
resulting power can be found by adding 1 to the exponent of the
original power. For example, 2
Again, notice each power that results. In this case, each
power can be found by adding 2 to the exponent of the original
power. For example, 2 This confirms our earlier observation that when we multiply two powers that have the same base, the exponent of the resulting power is the sum of the exponents in the two factors. Now is a good time to explore this idea for yourself. Choose
your own bases and exponents. Then evaluate both a
In general, when we multiply a
When we multiply a power of a times another power of a, the result is a power of a , where the exponent is the sum of the exponents of the two factors. In symbols, a This holds true for any number a and positive integers b and c .
## Laws of Exponents and Dividing MonomialsWhat happens when we divide powers? Let's analyze this by dividing various powers of 2.
Try to make a conjecture about dividing powers that have the same base. In both cases, the result is the original base raised to the power given by the difference of the two exponents. For 22 52 , 3 5 2, and for 22 73 , 4 7 3. The table on the left shows what happens when a power of 2 is
divided by 2
In the table on the left, notice the powers that result. Each resulting power can be found by subtracting 1 from the exponent of the original power. In the table on the right, each resulting power can be found by subtracting 2 from the exponent of the original power. This agrees with our original observation that when we divide two powers with the same base, the exponent of the resulting power is the difference of the exponents of the two dividends. Consider another case. Let's divide a
We can now cancel two a's from both numerator and denominator.
Cancellation is a shorthand process involving the properties of fractions. Also, point out that any number raised to the first power is that number itself.
After canceling, we find that aa 42 a 4 2 or a 2 . In general, when we write the quotient and expand it into products of a's, the result is
which shows why this fact is valid.
When we divide a power of a by another smaller power of a , the result is a power of a , in which the exponent is the difference of the exponents of the two dividends. In symbols, aa bc a b c when b c . This holds true for any nonzero number a and whole numbers b and c. |