19. Exponents and powers. You have learned that 4.4 may be written 42 and that a a may be written a2. SimiWe may larly 23 means 2.2.2 or 8, and a3 means a · a · a. indicate that 3 is used as a factor 5 times in either of the two ways 3.3.3.3.3 or 35. Which is the shorter way? Find the product. A power of a number is the product obtained by using the number as a factor one or more times. Thus, 34 or 3.3.3.3 or 81 is a power of the number 3. We read a2 as a square"; a3 as a cube"; and a1 as "a exponent 4" or "a fourth." In a4, 4 is called the exponent and a the base of the power. When the exponent is 1 it is usually omitted. Thus a1 is written a. = Since a2 means a a and a3 means a a· a, then a2. a3 = aaaaaa5. Similarly 35.32-3.3.3.3.3.3.3 37. The pupil should observe the difference between 2x3 and (2x)3. 2x3 means 2.x.xx, and (2x)3 means 2x-2x-2x =8x3. Rule. The product of powers having the same base equals that base with an exponent equal to the sum of the exponents of the powers. EXAMPLE 1. Find the value of 43. SOLUTION. 43=4.4.4 = 64. EXAMPLE 2. Multiply 24 by 23. SOLUTION. 24.23 = 27=128. EXAMPLE 3. Multiply a2b by a3c2b2. SOLUTION. a2b· a3c2b2 = a2 · a3 ⋅ b · b2 · c2 = a5b3c2. . Find the value of the following: 15. 24. 16. 33. 17. 52. 18. 73. 19. 45. 20. 27. 21. 103. 22. 104. 23. 105. 24. 108. 25. 210. 26. 55. 27. 253. Write these powers of 10, using exponents: 28. 100. 31. 1,000. 29. 10,000. 32. 1,000,000,000. 30. 1,000,000. 33. 100,000. 34. 104. 35. 53. Write the following without exponents, then multiply: 36. 122. 37. .14. 38. (.06)2. 20. Multiplying a polynomial by a monomial. 1. Using dots, show the sum of 5 dots and 7 dots. 2. With dots show 3 times the sum of 5 dots and 7 dots. 3. What is the meaning of 3(5+7)? 4. In this group each row contains (5+7) dots. The 3 rows contain 3(5+7) dots. Show that the left-hand group contains 3.5 dots ..... and the right-hand group contains 3.7 dots. ......... This shows that 3(5+7)=3.5+3.7. 5. Show with dots that 3(2+4+5)=3·2+3.4+3.5. These examples illustrate the following Rule. To multiply a polynomial by a monomial multiply each term of the polynomial by the monomial and add the products. 21. Multiplication by a polynomial. In multiplying 46 by 32 we get the products 306, 30-40, 26, and 2.40 and add them. Let the pupil show that the following are correct : 1. (2+3)(6+9)=2·6+2·9+3·6+3·9. = 2. 65.87 (60+5) (80+7)=60-80+60·7+5·80+5·7. 3. 48.391 (40+8) (300+90+1)= = 40 300+40.90+40·1+8.300+8.90+8.1. 4. (m+n)(x+y)=mx+my+nx+ny when m=3, n=8, x=4, y=12. FIG. 7 Such products can be illustrated by rectangles. Figure 7 shows that a rectangle 2+3 units wide and 4+5 units long contains 2.4+2.5+3.4+3.5 square units. The above examples illustrate the following Rule. To find the product of two polynomials multiply each term of one by each term of the other and add the products. EXAMPLE. Multiply 3m+5n by 2m+n. finding one of two factors when their product and the other factor are given. Thus, to divide 18 by 3 means to find the factor by which 3 must be multiplied to get 18. Then 18÷3=6, since 6.3=18. In the same way 12mn÷3m=4n, since 3m 4n Find what the first number must be multiplied by to get the Divide the first number by the second in the following: 23. Division of a polynomial by a monomial. In what two ways may 6. 4 be multiplied by 2 without first multiplying 6 by 4? Then in what two ways may 6. 8 be divided by 2 without first multiplying 6 by 8? How may 7+5 be multiplied by 3 without first adding 7 and 5? Then how may 21+15 be divided by 3 without first adding 21 and 15? Summary. To multiply, or to divide, a monomial by any number, multiply, or divide, any ONE of its factors by the number. To multiply, or to divide, a polynomial by any number, multiply, or divide, EACH term of the polynomial by the number. EXAMPLE. Divide 12x3y+8x2y2+6xy3 by 2xy. 1. Multiply 69 by 3 in three different ways. 2. Divide 18 36 by 6 in three different ways. 3. Divide 6. 14. 8 by 7 in the easiest way. |