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Multiplication and Division of Terms Study Guide (page 2)

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Updated on Oct 3, 2011

Example

(–12a3bc6)(–9xy9z5) =

Multiply the coefficients: (–12)(–9) = 108.

Put every base in either term into the answer. Our answer has bases of a, b, c, x, y, and z.

Because none of the bases in the first term appear in the second term, we have no exponents to add. The exponents of the bases in our answer are the same as the exponents of the bases in each term. Our answer is 108a3bc6xy9z5.

Dividing Terms

We follow some similar steps to divide terms. When multiplying two terms, we began by multiplying coefficients. To divide two terms, first, divide the coefficient of the first term by the coefficient of the second term. For bases that are in the dividend but not the divisor, carry those bases and their exponents into the answer. The next step might surprise you: If there are any bases in the divisor that are NOT in the dividend, carry them into our answer, too, but change the sign of their exponents. Finally, for each base that is common to both the dividend and the divisor, subtract the exponent of the base in the divisor from the exponent of the base in our dividend. That is the exponent of that base in our answer. Whew! Is it really that difficult? Not after you see a few examples.

Example

(8a5) ÷ (2a3) =

First, divide the coefficient of the dividend by the coefficient of the divisor: 8 ÷ 2 = 4.

Next, carry the bases of the dividend into your answer. Your answer has a base of a.

There are no bases in the dividend that are not in the divisor, and vice versa, so move right on to the next step.

For each base that is common to both the dividend and the divisor, subtract the exponent of the base in the divisor from the exponent of the base in the dividend. The exponent of a in 8a5 is 5 and the exponent of a in a3 is 3; 5 – 3 = 2. The exponent of a in our answer is 2. (8a5) ÷ (2a3) = 4a2.

Now, let's see an example where the dividend has a base that the divisor does not have, and the divisor has a base that the dividend does not have.

Example

(35g10) ÷ (5y4) =

Divide the coefficient of the dividend by the coefficient of the divisor: 35 ÷ 5 = 7.

Carry the bases of the dividend into the answer with its exponent, since that base is not present in the divisor. The answer has a base of g with an exponent of 10.

Carry the bases that are in the divisor but not the dividend into the answer. Change the sign of their exponents. The divisor has a base of y that is not in the dividend, so y will be in the answer, but with an exponent of –4 instead of 4.

There are no terms common to the dividend and the divisor, so we have no subtraction to do. (35g10) ÷ (5y4) = 7g10y–4.

Why did the exponent of y change from positive to negative? Because we wanted to divide the dividend by y, but there was no y in the dividend. That means we were unable to divide by y, and we need to show in our answer that this division was never performed. A negative exponent means that if we were to write a term as a fraction, the bases with negative exponents would be written in the denominator of the fraction (and their exponents would change to positive). Remember, fractions mean division, so if a term appears in the denominator of a fraction, that means the term is acting like a divisor. Our answer, 7g10y–4, could also be written as. In fact, we could write both the original problem and the answer as fractions:

Now, it's a little easier to see what happened: 35 was divided by 5,g10 was not divided by anything and went straight into our answer, and we could not divide by y4, so we kept it in the denominator of our answer.

You probably still have a few questions. What if a base is common to the dividend and the divisor, but the exponent of the divisor is greater? Or, what if a base is common to the dividend and the divisor, and their exponents are the same? Let's look at both of these cases.

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