Hence, the square root 1296 is 36; or, in other words 36 is the side of a square whose area is 1296. Hence we have CASE I. § 181. To extract the square root of a whole number. RULE. 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. Q. What is required when we wish to extract the square root of a number? What is the greatest square of a single figure? What is the highest order of units that can be derived from the square of a single figure? How many perfect squares are there among the numbers that are less than one hundred? What is the square of a number expressed by two figures equal to? In what places of figures will the square of the tens be found? In what places will the product of the tens by the units be found? 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. EXAMPLES. 1. What is the square root of 263169? 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 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. 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? 3. What is the square root of 106929? 4. What is the square root of 2268741 ? Ans. 327. Ans. 1506,23+. 5. What is the square root of 7596796 ? Ans. Ans. Ans. 4698. 6. What is the square root of 36372961? 7. What is the square root of 22071204? Q. How do you extract the square root of a whole number? How many figures will there be in the root? If the given number has not an exact root, what may be done? CASE II. § 182. To extract the square root of a decimal fraction.. RULE. I. Annex one cipher, if necessary, so that the number of decimal 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. Ans. 57,19+. 3. What is the square root of 4795,25731? Ans. 69,247+. 4. What is the square root of 4,372594 ? 5. What is the square root of ,00032754 ? Ans. 2,091+. 6. What is the square root of ,00103041? Ans. ,0321. 7. What is the square root of 4,426816? 8. What is the square root of 47,692836? Ans. Q. 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? CASE III. § 183. To extract the square root of a vulgar fraction. RULE. 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. Q. How do you extract the square root of a vulgar fraction? EXTRACTION OF THE CUBE ROOT. $ 184. 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. 2, 3, 4, 5, 6, 7, 8, 9. 8 27 64 125 216 343 512 729. 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, is 729,000; 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 one; 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. 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. 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 10×10×10=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. |