Multiplying Rational Numbers

You can only multiply two numbers at a time. Those two numbers have either like signs or unlike signs. From this information, we can develop our rules for multiplication.

Rules for Multiplication

• Like signs, positive
• Unlike signs, negative

Examples

 Reasoning 5(5) = 25   -5(-5) = 25 In both examples, the numbers have like signs. Both are either positive or negative. So the sign of the answer is positive. Reasoning 3(-5) = -15   -3(5) = -15 In both examples, the numbers have unlike signs. One is positive and the other is negative, so the sign of the answer is negative.

Remember: use your rules to find the sign of your answer first. Then do the indicated operation.

There is another situation that we need to address when multiplying. You can only multiply two numbers at a time, but we can multiply any number of numbers together by multiplying our first answer by the next number and so on, until we have multiplied all of the numbers together. Since the associative property says that we can multiply the numbers in any order, we can develop rules to first find the sign of our answer.

Multiplying more than two numbers

Count the number of minus signs in the problem

1. Even: positive
2. Odd: negative

Examples

 Reasoning -2(-8)(-4)   -64 Count the minus signs. There are three, which is odd. The answer is negative. Then multiply the numbers 2(8) = 16 and 16(4) = 64. Reasoning -2(4)(-5)   40 Count the minus signs. There are two, which is even. The answer is positive. Then multiply the numbers 2(5) = 10 and 10(4) = 40. Reasoning -2/3(-5)(6)(-4/5)   -2/3(-5/1)(6/1)(-4/5)   -2/3(-5/1)(26/1)(-4/5)   -2(1)(2)(4)   -16 I suggest that when you are working with fractions, change the integers into fractions by putting them over one. Then count the minus signs. There are three, which is odd, so your answer is negative. Then reduce where you can. Three goes into six twice and the fives cancel. Finally multiply 2(1) = 2, 2(2) = 4, 4(4) = 16

Multiplying expressions containing variables

You can also multiply expressions containing variables.

Examples:

 Reasoning -5a(6b)   -30ab Unlike signs, so your answer is negative. Multiply 5(6) = 30 and multiply a(b) = ab. Reasoning 3x(4y) - 2x(5y)   12xy - 10xy   2xy Remember: Order of operations tells us to multiply before we add.   Multiply 3x(4y), like signs; positive, 3x(4y) = 12xy   Multiply -2x(5y), Unlike signs; negative, 2x(5y) = 10xy   Add 12xy - 10xy = 2xy, Unlike signs; subtract and put the sign of the biggest number. In this example, remember that we can combine (add) like terms. Like terms Same variable(s) Same exponent(s) on those variable(s)

 Reasoning -4a(-3b) - 2a(7b)   12ab - 14ab   -2ab Multiply -4a(-3b) = 12ab, like signs; positive   Multiply -2a(7b) = -14ab, Unlike signs; negative   Add 12ab - 14ab = -2ab, Unlike signs; subtract and put the sign of the biggest number.

Just as a minus sign in front of parenthesis changes the sign of the number(s) inside the parenthesis so does multiplying by a negative one.

Examples

 Reasoning -1(5) = -5 Unlike signs; negative -1(-5) = 5 Like signs; positive This is called the Multiplication Property of -1. The product of any number and -1 changes the sign of the number (is its additive inverse). -1(a) = -a, a(-1) = -a

Multiplying Decimals

Now letâ€™s review the process of multiplying decimals. Remember: your answer must have the same number of decimal places as your factors.

Process for multiplying decimals

1. Multiply the numbers
2. Count the number of decimal places in your factors. That is, count how many numbers are to the right of the decimals.
3. In your answer, starting at the right count back that number of places and put your decimal there.
4. If there are not enough numbers, you must add zeros to the left of your number.

Examples

 Reasoning -.2(.7)   -.14 Unlike signs; negative   Multiply: 2(7) = 14   There are two decimal places so starting at the right of the four, we move two places to the left,  .14 Reasoning -.4(-.1)   .04 Like signs; positive   Multiply 4(1) = 4   There are two decimal places, so starting at the right of the four, we move two places to the left. But since there is only one number in our answer, we must add a zero to get .04

Try these exercises

Multiply

1. 3(4)
2. -5(7)
3. 6(-9)
4. -7(-8)
5. 2/3(-9/8)
6. -4/5(-15/7)
7. -3(-2)(-5)(-8)
8. 1/4(-8)(1/3)(-6)
9. .15(-3)
10. -.05(-.07)
11. Simplify

12. -2x(3y) - 4x(5y)
13. 3(2x - 6) - 4(3x - 4)
Hint: Use the distributive property.
14. -4(8) - 3(-5)
15. 2/3(6x - 9) + 3/4(8x -12)
Hint: Divide by the 3 or 4 then multiply by the 2 or 3.
16. -1.1(2.3) - .25(.3)

1. 3(4) = 12

2. -5(7) = -35

3. 6(-9) = -54

4. -7(-8) = 56

5. (2/3)(39/48) = -3/4

6. -4/5(-315/7) = 12/7

7. (-3(-2)(-5)(-8) = 240

8. (1/4)(-28/1)(1/3)(-26/1) = 4

9. .15(-3) = -.45

10. -.05(-.07) = .0035

11. -2x(3y) - 4x(5y)
-6xy - 20xy
-26xy

12. (2x - 6) - 4(3x - 4)
6x - 12 - 12x + 16
-6x + 4

13. 4(8) - 3(-5)
-32 + 15
-17

14. 1/3(6x - 9) + 3/4(8x - 12)
4x - 6 + 6x - 9
10x - 15

15. 1.1(2.3) - .25(.3)
-2.53 - .075
-2.605