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 25 101)131 101 1023)3069 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. 6. What is the square root of 36372961? 7. What is the square root of 22071204? Ans. Ans. 4698. 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? Ans. 2,091+. 5. What is the square root of ,00032754 ? 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. 1. 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. 2200 8, 9. Roots 1, 2, 3, 4, 5, 6, 7, Cubes 1 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 of 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 10x10x10=1000. 2. The second part 1800 is equal to three times the square of the tens multiplied by the units: that is 3x(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 3x62x10=3x36x10=1080. 4. The fourth part is equal to the cube of the units: that is 63=6×6×6=216. Let it now be required to extract the cube root of the number 4096. Since the number contains more than three figures, we know that the root will contain at least units and tens. 4 096(16 } 1 12+3=3)30 (9-8-7-6 162=4 096 Separating the three right hand figures from the 4, we know that the cube of the tens will be found in the 4. Now, 1 is the greatest cube in 4. Hence, we place the root 1 on the right, and this is the tens of the required root. We then cube 1 and subtract the result from 4, and to the remainder we bring down the first figure 0 of the next period. Now, we have seen that the second part of the cube of 16, viz., 1800 being three times the square of the tens multiplied by the units, will have no significant figure of a less denomination than hundreds, and consequently will make up a part of the 30 hundreds above. But this 30 hundreds also contains all the hundreds which come from the 3rd and 4th parts of the cube of 16. If this were not the case, the 30 hundreds divided by three times the square of the tens would give the unit figure exactly. Forming a divisor of three times the square of the tens we find the quotient figure to be ten-but this we know to be too large. Placing 9 in the root and cubing 19, we find the result to be 6859. Then trying 8 we find the cube of 18 still too large-but when we take 6 we find the exact number. Hence the cube root of 4096 is 16. |