Then, placing BC or 6, in the root, and also in the divisor, and then multiplying the whole divisor 66 by 6, we obtain for a product the two rectangles FE and CE, together with the square ED. Hence, the square root of 1296 is 36; or, in other words, 36 is the side of a square whose area is 1296. CASE I. 308. To extract the square root of a whole number, I. Point off the given number into periods of two figures each, counted from the right, by setting a dot over the place of units, another over the place of hundreds, and so on. II. Find the greatest square in the first period on the left, and place its root on the right after the manner of a quotient in division. Subtract the square of the root from the first period, and to the remainder bring down the second period for a dividend. III. Double the root already found and place it on the left for a divisor. Seek how many times the divisor is contained in the dividend, exclusive of the right hand figure, and place the figure in the root and also at the right of the divisor. IV. Multiply the divisor, thus augmented, by the last figure of the root, and subtract the product from the dividend, and to the remainder bring down the next period for a new dividend. But if the product should exceed the dividend, diminish the last figure of the root. V. Double the whole root already found, for a new divisor, and continue the operation as before, until all the periods are brought down. EXAMPLES. 1. What is the square root of 263169 ? QUEST.-308. What is the first step in extracting the square root of numbers? What the second? What the third? What the fourth? What the fifth? Give the entire rule. We first place a dot over the 9, making the right hand period 69. We then put a dot over the 1 and also over the 6, making three periods. OPERATION. 26 31 69(513 25 101)131 101 1023)3069 The greatest perfect square in 26, is 25, the root of which is 5. Placing 5 in the root, subtracting its square from 26, and bringing down the next period 31, we have 131 for a dividend, and by doubling the root we have 10 for a divisor. Now 10 is contained in 13, 1 time. Place 1 both in the root and in the divisor: then multiply 101 by 1; subtract the product and bring down the next period. We must now double the whole root 51 for a new divisor, or we may take the first divisor after having doubled the last figure 1; then dividing we obtain 3, the third figure of the root. 309. NOTE 1.-There will be as many figures in the root as there are periods in the given number. NOTE 2.-If the given number has not an exact root, there will be a remainder after all the periods are brought down, in which case ciphers may be annexed, forming new periods, each of which will give one decimal place in the root. 2. What is the square root of 36729? QUEST.--309. How many figures will there be in the root? If the given number has not an exact root, what may be done? 3. What is the square root of 213444 ? CASE II. Ans. Ans. Ans. Ans. Ans. 310. To extract the square root of a decimal fraction, I. Annex one cipher, if necessary, so that the number of decımal places shall be even. II. Point off the decimals into periods of two figures each, by putting a point over the place of hundredths, a second over the place of ten thousandths, &c.: then extract the root as in whole numbers, recollecting that the number of decimal places in the root will be equal to the number of periods in the given decimal. NOTE. When there is a decimal and a whole number joined together the same rule will apply. 2. What is the square root of 3271.4207 ? Ans. QUEST.-310. How do you extract the square root of a decimal fraction? When there is a decimal and a whole number joined together, will the same rule apply? 6. What is the square root of .00103041 ? 7. What is the square root of 4.426816 ? Ans. Ans. 8. What is the square root of 47.692836 ? Ans. CASE III. 311. To extract the square root of a vulgar fraction, I. Reduce mixed numbers to improper fractions, and compound fractions to simple ones, and then reduce the fraction to its lowest terms. II. Extract the square root of the numerator and denominator separately, if they have exact roots; but when they have not, reduce the fraction to a decimal and extract the root as in Case II. EXAMPLES. 1. What is the square root of 23 of 5 of 41? 312. To extract the cube root of a number is to find a second number which being multiplied into itself twice, shall produce the given number. Thus, 2 is the cube root of 8; for, 2 × 2 × 2 = 8: and 3 is the cube root of 27; for, 3 × 3 × 3 = 27. QUEST.-311. How do you extract the square root of a vulgar frac From which we see, that the cube of units will not give a higher order than hundreds. We may also remark, that the cube of one ten or 10, is 1000: and the cube of 9 tens or 90, 729000; and hence, the cube of tens will not give a lower denomination than thousands, nor a higher denomination than hundreds of thousands. Hence also, if a number contains more than three figures its cube root will contain more than if the number contains more than six figures the root will contain more than two; and so on, every three figures from the right giving one additional place in the root, and the figures which remain at the left hand, although less than three, will also give one place in the root. one; Let us now see how the cube of any number, as 16, is formed. Sixteen is composed of 1 ten and 6 units, and may be written 10+6. Now to find the cube of 16 or of 10 +6, we must multiply the number by itself twice. To do this we place the numbers thus 10+ 6 1. By examining the composition of this number it will be found that the first part 1000 is the cube of the tens; that is, 10 x 10 x 10 — 1000. 2. The second part 1800 is equal to three times the square of the tens multiplied by the units; that is, 3 × (10)2 × 6 = 3 × 100 × 6 = 1800. 3. The third part 1080 is equal to three times the square of the units multiplied by the tens; that is, 3 × 62 × 10 = 3 × 36 × 10 = 1080. |