Rule for finding the two numbers when given the first two terms of the square of their sum. Find the first number by taking the square root of the first term given. To find the second number divide the second term given by twice the first number, which has just been found. 26. 400+40+(?)2= (?+?)2. 27. 100+160+(?)2=(?+?)2. 28. 900+240+(?)2=(?+?)2. 29. 3600+840+(?)2=(?+?)2. 30. 520+(?)2=400+120+(?)2= (?+?)2. 31. 1140+(?)2 = (?+?)2. HINT. The largest number of hundreds in 1140 which is a square number is 900. Separate 1140 into the two terms 900+240. In the following, first separate the given number into two terms so that the first shall be the largest number of hundreds which is a square. 32. 560+(?)2= (?+?)2. 33. 2800+(?)2= (?+?)2. 34. 280+(?)2= (?+?)2. 35. 1200+(?)2= (?+?)2. 36. 680+(?)=(?+?)2. 37. 3200+(?)2 = (?+?)2. 38. 1840+(?)2 = (?+?)2. In finding the square root of any number we separate it into three terms as in the expression a2+2ab+b2, whose square root we know how to find. Sometimes the first term is given separate but the last two terms are given combined. Thus, 400+129=(??)2. In this case we do not know how much of 129 is 2ab and how much is b2. However, if a is a large number compared with b, then 2ab is much larger than b2, and we can find b approximately by dividing all of 129 by 2a, that is by 2 20. Dividing 129 by 40 gives 3 approximately. Then 2ab+b2-220.3+32=129. And 400+129 (20+3)2=400+120+9. Supply the missing numbers in the following : 39. 4900+284=(?+?)2. 40. 1600+81=(?+?)2. 41. 6400+489 = (?+?)2. 42. 2500+416=(?+?)2. 43. 900+396 (?+?)2. = 44. 400+384=(??)2. 45. 1600+704=(?+?)2. 46. 3600+1161=(??)2. 28. The number of orders in the square of a number. To find the square root of a number like 1849 it is necessary to learn how to separate it into terms so that the plan of the preceding exercise may be followed. When squaring a number such as 43 we really separate it into 40+3, that is, into its tens and units. We must now study the number of orders in the squares of the different numbers. Fill the blanks in the following: How many orders in the numbers 1 and 9? In 10 and 99? In 100 and 999? How many orders in the squares of numbers of one order? How many orders in the squares of numbers of two orders? In the squares of numbers of three orders? These examples illustrate the Principle. The number of orders in the square of a number is twice as many as in the number, or one less than twice as many. It follows from this rule that if a square number contains an even number of orders, its square root contains one-half as many orders. If a square number contains an odd number of orders add 1 to the number of orders and divide the sum by 2, to get the number of orders in the square root. Thus the square root of 16,777,216 contains 4 orders, and the square root of 27,225 contains 3 orders. FINDING SQUARES OF UNITS, TENS, HUNDREDS 51 Exercise 30 1. Tell how many orders there are in the square of each of the following numbers: 8, 3, 40, 98, 200, 7, 46, 345. 2. Give the square root of each of the following: 9, 49, 64, 81. 3. If a number contains one or two orders, how many orders are there in its square root? 4. Give the square root of each of the following: 400, 225, 900, 625, 100, 6400, 8100, 4225. How many orders in each of these numbers? How many orders in the square root of each? 5. Without finding the square roots of the following numbers tell how many orders there are in each square root: 289, 841, 6084, 4, 36, 7921, 27889, 35344, 148996, 606841. 29. Finding the squares of units, tens, and hundreds. Let the pupil fill the blanks in the following: 32= 82= 302= = 3002= These examples illustrate the fact that the square of units' digit is found in units' and tens' orders, the square of tens' digit is found in hundreds' and thousands' orders, and that the square of hundreds' digit is found in ten-thousands' and hundred thousands' orders. In finding the square roots of numbers we shall first want to know how many orders there are in the square root. Thus, if we want to find the square root of 86436, we first see that the square root contains three orders. Why? What are the names of these orders? We know that the square of the number of hundreds in this root is found in tenthousands. The largest square in 8 is 4. The number of hundreds in the square root is then 2. The places of the squares of the orders of the root are usually indicated by pointing off the number whose square root is to be found into periods of two figures each, beginning at the right, thus, 8'64′36. 30. To find the square root of any integer. We wish to find the square root of 1849. We point off 1849 into periods of two figures each and find that there will be two orders in the root. The square of the tens must be found in the 1800, and the largest number of tens whose square is found in 1800 is 4 tens, or 40. Its square is 1600. We then separate 1849 into 1600+249 and finish the work as in example 39 of exercise 29. Exercise 31 Find the square root of each of the following: 1. 1089. 2. 3844. 3. 5625. 4. 2809. 5. 5329. 6. 784. 7. 3136. 8. 8464. 9. 1521. 10. 256. The pupil will now find convenient the following form for computing square roots. 11. Find the square root of 3969. 3 is added to 120 and the sum multiplied by 3. In practice this form is much condensed. Find the greatest square in 39 to be 36. 39'69)63 36 Place the square root of 36 as the first figure in 123 369 the root. Subtract the 36 and bring down the next two figures. 369 For trial divisor use 2.6. Divide the remainder, omitting the final digit 9, by the trial divisor. 36÷12=3. Annex this figure to the root, also to the trial divisor. Multiply this complete divisor by the new figure of the root. 12. Find the square root of 57121. SOLUTION. periods. Separate the number into Find the greatest square in 5 and subtract it. Divide 17 by the trial divisor 2 · 2. Annex the quotient 4 to the trial divisor, also to the figure of the root already found. 5'71'21)239 4 43 171 129 469 4221 4221 Multiply the complete divisor by the new figure of the root. The product is more than 171, proving that 4 is too large. Try 3. For the next trial divisor double the part of the root already found, 23, and divide 422 by the product. Annex the quotient 9 to the trial divisor and also to the root. Multiply the complete divisor by the new figure of the root and subtract as before. Find the square root of each of the following: 13. 2209. 16. 46225. 22. 516961. 19. 94249. 14. 54756. 17. 795664. 15. 58081. 20. 39204. 23. 401956. 18. 182329. 21. 804609. 24. 11641744. 31. To find the square root of a decimal fraction. Fill the blanks in the following: .12= .92= .012= .0012= .0092 = The square of a number having one decimal order has how many decimal orders? Of a number having two decimal orders? Of a number having any given number of decimal orders? The pupil has seen that the square of any number of tenths is a number of hundredths; the square of a number of hundredths is a number of ten-thousandths, etc. Therefore to find the square root of a decimal fraction begin at the decimal point and separate it into periods of two figures each. The square root of the greatest square in the first period to the right of the decimal point is the tenths' figure of the root. The other digits are found as for integers. If there is an odd number of decimal orders annex a zero. |