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Pre Algebra Section 1.3 Unit 1 - Integers Section 3 – Multiplying and Dividing Integers MULTIPLICATION Consider the product 3 ∙ −5 Remember that means that you have three (-5) added together, in other words (−5 ) + (−5 ) + (−5) = −10 + −5 = −15 We know that 3 ∙ 5 = 15 . From the above we can see that if one number is a negative the answer will become negative. Since multiplication in commutative (we can change the order and get the same result) −5 ∙ 3 = -15 as well. When you multiply two numbers that have opposite signs, the result is negative. Remember that a negative sign means opposite. Therefore if you have two negatives you can think of it as taking the opposite of a number and then take the opposite of that which leaves you with the original number. For example −2 ∙ −5 = 10 When you multiply two numbers that have the same sign, the result is positive. One fun way to remember that multiplying two negatives makes a positive is to think of one of the negative signs turning vertical to combine with the second negative to make the plus sign. RULE: When you multiply two numbers that have opposite signs, the results is negative. When you multiply two numbers with the same sign, the result is positive. American River College 25 Milano Pre Algebra Section 1.3 Example 1) ࡹ࢛࢚࢟ ૡ ∙ ૠ 8∙ 7 = 56 Same sign so the answer is positive Example 2) ࡹ࢛࢚࢟ − ∙ ૡ −4 ∙ 8 = −32 Opposite signs so the answer is negative Example 3) ࡹ࢛࢚࢟ ૢ ∙ − 9 ∙ −4 = −36 Opposite signs so the answer is negative Example 4) ࡹ࢛࢚࢟ − ∙ −ૠ −6 ∙ −7 = 42 American River College Same sign so the answer is positive 26 Milano Pre Algebra Section 1.3 Properties of Multiplication Let ܽ be any real number, then the following properties are true. The Multiplication Property of Zero ܽ ∙ 0 = 0 The Multiplication Property of One ܽ∙1 = ܽ The Associative Property of Multiplication (ܽ ∙ ܾ) ∙ ܿ = ܽ ∙ (ܾ ∙ ܿ) The Commutative Property of Multiplication ܽ∙ܾ = ܾ∙ܽ So what do you do if there are more than 2 numbers? The following is an example of that. Example 5) (−)()(−) We will start with the first two numbers 3(-2) which means 3 ∙ −2, which is -6, so the problems changes from 3(−2)(6)(−4) = −6 (6)(−4) Now combine the next two = −36 (−4) and the last two = 144 is the final answer American River College 27 Milano Pre Algebra Section 1.3 Evaluating expressions with Multiplication We can also evaluate expressions that contain multiplication. Example 6) Evaluate ࢞ , for ࢞ = −ૢ 4 ݔmeans 4 ∙ ݔ Therefore replacing ݔwith the −9 gives 4∙ݔ = 4 ∙ −9 = − 36 which is the final answer. DIVISION Every Division problem can be rewritten as a multiplication problem. Therefore the rules of multiplication can be extended to division as well. Notice the similarities between the following −2 ∙ 3 = −6 and −6 ÷ 3 = −2 We can even goes as far as saying −6 ÷ 3 = −2 because −2 ∙ 3 = −6. Therefore it makes sense the rules will be the same as well. RULE: When you Divide two numbers that have opposite signs, the results is negative. When you Divide two numbers with the same sign, the result is positive. Example7) − ÷ −55 ÷ 5 = −11 American River College 28 Milano Pre Algebra Section 1.3 Example 8) −ૠ ÷ −ૡ −72 ÷ −8 = 9 Example 9) ଵସଶ ି ିଶ is the same as 142 ÷ −2 142 ÷ −2 = − 71 Example 10) ିૡ ିૢ ି଼ଵ ିଽ =9 Properties of Division When dividing it is important to remember you cannot divide by zero! • • • ଵ ݂݅݀݁݊݅݁݀݊ݑ ݏ =ܽ ݂݅ ܽ ≠ 0, ݐℎ݁݊ = 1 ܽ݊݀ =0 Evaluating expressions with division We can evaluate expressions that contain division in the same way we have evaluated other expressions. Example 11 ) ࢞ ࡱ࢜ࢇ࢛ࢇ࢚ࢋ , ࢌ࢘ ࢞ = − ࢇࢊ ࢟ = . ࢟ ௫ Replacing x with -125 and y with 5 gives ௬ = ିଵଶହ ହ = −25 American River College 29 Milano Pre Algebra Section 1.3 Area of a rectangle The area of a rectangle can be found by multiplying the base times the height. b h ∙ ܾ = ܣℎ or ݓ ∙ ݈ = ܣ Example 12) Find the area of a rectangle whose bas is 10 cm and Height is 3 cm. 5cm 4cm ∙ܾ=ܣℎ = ܣ5ܿ݉ ∙ 4 ܿ݉ = ܣ20ܿ݉ ଶ The area in 20 square cm. American River College 30 Milano Pre Algebra Section 1.3 Exercise 1.3 NAME:_________________________________ Multiply 1. 3(−4) 2. −4 ∙ 6 3. -4 ∙ 7 4. −3( −5) 5. −8(−8) 6. −11(−7) 7. −8 ∙ 3(−2) 8. 6 ∙ (−2) ∙ 5 9. (−5)(−6)(−2) 10. Find the product of 8 and -13. 11. Find the product of -7 and – 4 12. Find the product of -6 and 9 13. ݕݔ ݁ݐܽݑ݈ܽݒܧ, ݂ = ݔ ݎ−6 ܽ݊݀ = ݕ9 14. ݁ݐܽݑ݈ܽݒܧ6ܾܽ, ݂ = ܽ ݎ−3 ܽ݊݀ ܾ = −7 15. Evaluate −7ݕݔ, ݂ = ݔ ݎ−1 ܽ݊݀ = ݕ6 American River College 31 Milano Pre Algebra Section 1.3 Dividing 16. −33 ÷ 3 17. −95 ÷ −5 ିଷ 19. ିଵଶ 22. Evaluate 20. ି 18. 240 ÷ −12 ିଶ଼ 21. ସ ି଼ , ݂ = ܽ ݎ72, ܽ݊݀ ܾ = −4. 23. Evaluate , ݂ = ܽ ݎ22 ܽ݊݀ ܾ = −2. 24. Evaluate ܽ ÷ ܾ, ݂ = ܽ ݎ56, ܽ݊݀ ܾ = 8. For problem 25-27, consider the following rectangle. h b Find the area of the rectangle with dimensions below. 25. ܾ = 3݉, ℎ = 4݉ American River College 26.ܾ = 7݂ݐ, ℎ = 5݂ݐ 32 27. ܾ = 11ܿ݉, ℎ = 4ܿ݉ Milano