Multiplication and Division of Terms Study Guide (page 3)
Introduction to Multiplication and Division of Terms
Can you do Division? Divide a loaf by a knife—what's the answer to that?
—Lewis Carroll (1832–1898) English Author and Mathematician
In this lesson, you'll learn how to multiply and divide like and unlike terms.
Now that you know how to add and subtract terms, let's look at how to multiply and divide them. While unlike terms cannot be combined (added or subtracted), we can multiply and divide both like terms and unlike terms. In fact, multiplication and division of unlike terms is no different from multiplication and division of like terms!
Just as with addition and subtraction though, we must first review signed numbers. What happens when a positive factor is multiplied by a negative factor? Or when two negative factors are multiplied? The following chart might look familiar.
When two factors of the same sign are multiplied, the product is positive. A product is the result when you are multiplying two factors. When two factors of different signs are multiplied, the result is negative.
Always remember: Two of the same sign, think positive; two different signs, think negative.
The division chart looks just like the multiplication chart.
In a division problem, the term being divided is called the dividend, the term doing the dividing is called the divisor, and the result of the division is called the quotient.
When we added two like terms, we worked only with the coefficients. The base and exponent of our sum was the same as the base and exponent of each addend (an addend is a term that is added to another term). To multiply two terms, we must work with the coefficients, the bases, and the exponents.
First, multiply the coefficients. This product is the coefficient of our answer. Next, put every base in either term into the base of our answer. Finally, add the exponents of the bases that each term has in common, and those sums are the exponents of each base in the answer. Confused? Let's look at an example.
What is the product of 2m3 and 5m5?
First, multiply the coefficients: (2)(5) = 10.
Next, put every base in either term into the answer. Both terms have a base of m, so the base of our answer is m.
Finally, add the exponents of the bases that each term has in common. In other words, add the exponent of m in 2m3 to the exponent of m in 5m5.
The exponent of m in 2m3 is 3 and the exponent of m in 5m5 is 5. 3 + 5 = 8.
The exponent of m in our answer is 8.
Now, put it all together. The coefficient of our answer is 10, the base of our answer is m, and the exponent of our answer is 8:10m8. The product of 2m3 and m5 is 10m8.
Multiplying can be a bit trickier when the bases of each factor are a little different. Here's another example.
Follow the same steps. First, multiply the coefficients: (8)(–9) = –72.
Put every base in either term into the answer. Both terms have a base of x, so x is a base in our answer. The irst term also has a base of y, so our answer has a base of y, too.
Next, add the exponents of the bases that each term has in common. The exponent of x in 8x2y3 is 2 and the exponent of x in –9x7 is 7, so the exponent of x in our answer is 2 + 7 = 9. The exponent of y in 8x2y3 is 3.
y does not appear at all in the second term, so the exponent of y in our answer is 3.
Finally, put it all together: the coefficient of our answer is –72, one base of our answer is x with an exponent of 9, and the other base of our answer is y with an exponent of 3: –72x9y3.
Multiplying terms with completely different bases is actually easiest of all. Because the terms have no bases in common, there are no exponents to add. Just copy the bases and exponents of each term right into your answer.
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.
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.
(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.
(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.
(18a4b3) ÷ (–6ab5) =
Begin with the coefficients: 18 ÷ –6 = –3.
Carry the bases of the dividend into the answer. The answer has bases of a and b.
There are no bases in the dividend that are not in the divisor, and vice versa, so we are ready to subtract.
The exponent of a in the dividend is 4 and the exponent of a in the divisor is 1: 4 –1 = 3. The exponent of a in the answer is 3. The exponent of b in the dividend is 3 and the exponent of b in the divisor is 5. 3 –5 = –2. The exponent of b in the answer is –2.
Put it all together: (18a4b3) ÷ (–6ab5) = –3a3b–2. It is okay to have a negative exponent in your answer! If you would like to avoid negative exponents, this answer can also be written as.
(16c7) ÷ (4c7) =
Begin again with the coefficients: 16 ÷ 4 = 4.
There is only one base in the dividend and the divisor, c. Subtract the exponent of c in the dividend from the exponent of c in the denominator: 7 – 7 = 0. What does this mean? We could write our answer as 4c°, but because any number or variable to the power of 0 is equal to 1, c° = 1 and 4(1) = 4. In other words, if the exponent of a base is the same in both the dividend and the divisor, that base does not appear in your answer.
If the coefficient of the dividend and the divisor are the same, your answer does not appear to have a coefficient. That is because any value divided by itself is equal to 1, and it is unnecessary to write the number 1 as the coefficient of a term.
Find practice problems and solutions for these concepts at Multiplication and Division of Terms Practice Questions.
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