Elementary Algebra and Geometry for Schools

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 HOME : Mathematics - Geometry The product of three or more numbers. As you learned in arithmetic, if three or more numbers are to be multiplied together, it does not matter in which order the numbers are multiplied. That is, to multiply 2 X 3 X 4, we might first multiply the first two numbers together and then multiply that result by the third number, as: We might first multiply the last two numbers together and then multiply that result by the first number, as: We might first multiply the first and third numbers together and then multiply that result by the second number, as: When some of the numbers are positive and some are negative, we may follow the same method as that used above. Since the product of two negative numbers is positive, whenever there is an even number (two, four, six, etc.) of negative factors in a product, the product is positive; whenever there is an odd number (one, three, five, etc.) of negative factors in a product, the product is negative. ILLUSTRATIVE EXAMPLES: Finding the product of three or more numbers 1. Find the product: Solution. Since this term contains an odd number (three) of negative factors, the product will be negative. Multiplying the four factors together, and labeling the product minus, we have: 2. Simplify: Solution. Since this term contains four (an even number of) negative factors, the answer will be positive. Hence, we may proceed to simplify the term  by canceling common factors in the numerator and the denominator; that is, we may divide the 20 in the numerator and the 35 in the denominator by the common factor 5, and we may divide the 18 in the numerator and the 66 in the denominator by the common factor 6. This work is usually shown as follows: Multiplying the factors in the numerator and in the denominator = 144/77 3. Multiply: (4a2b3c3)(-3a2b3c2). EXERCISES: Multiplying and dividing signed numbers Simplify each of the following: Answers : Powers of negative numbers. We learned before that a3 means a . a . a, and that 34 means 3 .3 . 3 . 3. Similarly, (-a)3 means .(-a)(-a)(-a), and (-3)4 means (-3)(-3)(-3)(-3). Thus, if a negative number is raised to an even power, it contains an even number of negative factors; hence its value will be positive. If a negative number is raised to an odd power, it contains an odd number of negative factors; hence its value will be negative. We should, at this point, notice the difference between -a2 and (-a)2. The notation -a2 means -1a2, and the -1 is not raised to the second power. Hence, -a2= 1a2, while (-a)2 means (-a)(-a) and is equal to a2. Similarly, -24 means -1(2)4 which is equal to -16, while (-2)4 is equal to 16. ILLUSTRATIVE EXAMPLES: Evaluating powers of negative numbers 1. Find the value of (-3)5. Solution. Since the negative number -3 is raised to an odd power, its value will be negative; hence (-3)5 = -243 2. Find the value of -2(-4)3. Solution. Since the negative number -4 is raised to an odd power, its value will be negative; hence -2(-4)3 = -2(-64)              = 128 ORAL EXERCISES: Finding the values of powers of negative numbers Give the value of each of the following: Answers : 1. 9. 2. 16. 3. -125. 4. -2. 5. -27. 6. -5. 7.-3. 8. 1. 9. 256. 10. -1. 11. -32. 12. 25. 13. -64 14. 625. 15. -3125. 16. 6. 17. -1. 18. -8. 19. -4. 20. 4. 21. 1 22. -1024. 23. -243. 24. 81. 25. -1. Evaluating algebraic expressions. In other pages we evaluated algebraic expressions when we were given the numerical values of the letters. In those problems the values of the letters were positive. We will now evaluate expressions when the values of the letters are signed numbers; that is, when the values of the letters are positive or negative. ILLUSTRATIVE EXAMPLES: Evaluating algebraic expressions ORAL EXERCISES: Evaluating algebraic expressions 1. If x =-2, what is the value of 3x ? 2. If a =-5, what is the value of -2a ? 3. If y =-1, what is the value of -y ? 4. If b =-3, what is the value of 9 +b ? 6. If b =-3, what is the value of 9 -b ? 6. If x =-5, what is the value of 9 -2x ? 7. If y =-6, what is the value of 2y + 12 ? 8. If x =-4, what is the value of 3x -7 ? 9. If a =-2, what is the value of a2 ? 10. If b =-3, what is the value of -2b2 ? 11. If x =-3, what is the value of 4 + x2 ? 12. If y =-2, what is the value of 6 -y2 ? Answers : 1. -6. 2. 10. 3. 1. 4. 6. 5. 12. 6. 19. 7. 0. 8. -19. 9. 4. 10. -18. 11. 13. 12. 2. EXERCISES: Evaluating algebraic expressions If a =-2, b = 3, x = -1, and y = -3, find the value of each of the fallowing expressions: Answers : 1. 4. 2. -6. 3. -7. 4. 11. 6.9. 6. 7. Answers : 7. -32. 8. -1. 9. 5. 10. 7. 11. 10. 12. -8. 13. 13. 14. 5 15. 16. Find the numerical value of each of the following: Answers : 16. (a) -8; (b) 8; (c) 8; (d) 24. 17. (a) 9; (b) -27; (c) -9; (d) -405. 18. (a) -64; (b) 256; (c) -48; (d) 512. 19. (a) -4; (b) 2; (c) -4; (d) 16. 20. (a) -75; (b) 90; (c) 675; (d) 135. CHAPTER REVIEW EXERCISES Answers : 1.-3. 2. -18. 3.6. 4. -5a2. 5. -6xy2. 6. 3x2. 7.20. 8. -3y. 9. -6a3b3. 10. 18. 11. -8. 12.0. 13. -7. 14. 12. 15. -24. 16. 12. 17. -50. 18. 8. 19. 24a2b2. 20. 4ax -12ay -2xy. 2a(3x -4y) -x(2a +3y) -(4ay -xy) Answers : 21. 2. 22. (3x -y) -(2a -b). 23. 36a3b2e3. 24. 2(a + b). 25. x -(a + b). 26. -300. 27. 3x2 -10x -3. 28. -4. 29. -9. 30. -15. CUMULATIVE REVIEW EXERCISES 1. In the expression 7x, what is the coefficient of x? 2. Express algebraically, "four less than twice x." 3. Find the value of 3xy when x = 2 and y =-4. 4. Multiply x2 by x3 5. If 3x = 18, find the value of x. 6. Divide 6xy by 3x. 7. Solve for x: 5x -7 = 13. 8. Twice a certain number, when increased by 11, is equal to 23. Find the number. 9. Subtract 45 from 27. 10. If x =-2, find the value of 2x + 5. 11. Find the value of: (2)(-5)(-3). 12. Find the quotient: -12/-3 13. What is the value of (-3)3? 14. Simplify by collecting like terms: 3y -5y -2y + y. 15. Divide -10 by -¼. 16. In the expression 7x2y, what is the exponent of x? the exponent of y? 17. Find the value of the expression 3x2 + 4x + 7 when x =-2. 18. Solve the equation 9x = 45. 19. Solve the equation 7x -9 = 5 20. Solve the equation 5x -2x = 19 + 8. Answers : 1. 7. 2. 2x -4. 3.-24. 4. x5. 5. 6. 6. 2y. 7. 4. 8. 6. 9. -18. 10. 1. 11. 30. 12. 4. 13. -27. 14. -3y. 15. 40. 16. 2,1. 17. 11. 18. 5. 19. 2. 20. 9.