1. Point off the integer into periods of two figures each, beginning at the right. 2. Find the greatest perfect square that is not greater than the left-hand period. Subtract it from the left-hand period and write its square root at the right of the given integer for the first figure of the root. 3. Bring down the next period. 4. Multiply the part of the root already found (assuming that a cipher is annexed), by 2, and write the product at the left of the remainder for a trial divisor. 5. Divide the remainder (with period annexed) by the trial divisor. Write the quotient in the root, and also annex it to the trial divisor, making the divisor complete. 6. Multiply the complete divisor by the new figure in the root. Subtract the product from the last remainder (with period annexed) and proceed as before until all the periods of the square have been used. 7. When the remainder (with period annexed) will not contain the trial divisor, place a cipher in the root, bring down another period, and annex a cipher to the trial divisor for a new trial divisor. 1. 8836 6. 60,025 11. 235,225 16. 792,100 12. 16,184,529 17. 30,250,000 13. 5,322,249 18. 64,480,900 19. 43,560,000 20. 49,084,036 552. Oral THE SQUARE ROOT OF A DECIMAL 1. Find the square of .2; .3; .8; .9; .01; .05; .07; .12; .08; .001; .005; .011; .008. 2. When we square a decimal of one place, how many decimal places do we obtain in the square? Of two places? Of three places? Of four places? 3. The number of decimal places in the square compares how with the number of decimal places in its square root? 4. The number of decimal places in the root compares how with the number of decimal places in its square? 5. Can a perfect square have one decimal place? Three decimal places? Seven decimal places? Five decimal places? 6. Can any number be multiplied by itself so as to obtain a number consisting only of a figure in units' place and a figure in tenths' place? 553. The above discussion forms the basis of the following Summary To find the square root of a decimal: 1. Beginning at the decimal point, point off the decimal, both to the left (in a mixed decimal) and to the right, into periods of two figures each. 2. Find the square root as with integers. 3. Point off one decimal place in the root for every two decimal places in the square. NOTE 1. - If the given decimal contains an odd number of decimal places, a cipher must be annexed to complete the right-hand period. NOTE 2. The square root of a decimal or an integer that is not a perfect square may be found correct to any desired number of decimal places by annexing decimal periods of ciphers and continuing the work of extracting the square root. 2. Find, correct to two decimal places, the square root of: From the above illustration, tell how a may be squared. How may we find the square root of 49? Of 25? Of ? To find the square root of a common fraction: 1. Reduce the given fraction to lowest terms. 2. Extract the square root of the numerator and of the denomi nator. 3. If either numerator or denominator is not perfect square, change the common fraction to a decimal and find the square root correct to the required number of decimal places. To find the square root of a mixed number: 1. Change the mixed number to an improper fraction. 2. Find the square root by the method given above. 556. Oral 25 64 16 63 36 700 1. Find the square root of:;; §1; 12; 18; 7%; 21; 28; 13; 25; 11%. 558. The square root of a perfect square, the cube root of a perfect cube, or any root of the corresponding perfect power may be found by factoring. To determine the method of evolution by factoring, and the reason for it, let us study the relation between the factors of a number and the factors of the square of that number. 42 = 2 × 3 × 7; therefore 422 = (2 × 3 × 7)2 = 2 × 3 × 7 × 2 × 3 × 7, or 1764. We observe that every factor of 42 occurs twice in the square of 42. Likewise, every factor of any number occurs twice in the square of that number, three times in its cube, four times in its fourth power, and so on. 42. Conversely, 1764 = √2 × 2x3x3x7x7 = 2 × 3 × 7, or Likewise √225 = √ 3 × 3 × 5 × 5 = 3 × 5, or 15. √216 = √2 × 2 × 2 × 3 × 3 × 3 = 2 × 3, or 6. Summary 1. The square root of a perfect square may be found by factoring the square and multiplying together one out of every pair of equal prime factors found in it. 2. The cube root of a perfect cube may be found by factoring the cube and multiplying together one of every three equal prime factors found in it. |