SQUARE ROOT OF FRACTIONS. 624. The square root of a common fraction is the square root of its numerator divided by the square root of its denominator. Hence, when both terms are perfect squares, we have the following RULE.-Extract the square root of each term separately; the results will be the corresponding terms of the fraction which is the required root. Or, reduce the given fraction to a decimal, and then extract its root by the previous rule. Unless both terms of a fraction are perfect squares, the fraction is not a perfect square, and only its approximate root can be found. Either term, however, can always be made a perfect square, without altering the value of the fraction, by multiplying both terms by the same APPROXIMATE SQUARE ROOT OF FRACTIONS. 625. To extract the square root of a fraction which is not a perfect square, to within less than its fractional unit. RULE.-Multiply the numerator by the denominator, and extract the square root of the product to within less than 1; divide this result by the denominator of the given fraction; the quotient will be the root required. NOTE. This is equivalent to multiplying both terms by the same number, and extracting the root of both. EXAMPLES. 1. Extract the square root of to within less than. and the square root of 70, within less than 1, is 8; hence or is the required approximate root. 2. Find the square root of to within less than 1. Ans. †. 3. Extract the square root of to within less than NOTE. To make a nearer approximation to the true root, increase the denominator to any larger square. 626. To extract the square root of a fraction which is not a perfect square, to within less than any fractional unit. RULE. Reduce the given fraction to the same denomination as the fractional unit, and then proceed according to the previous rule. EXAMPLES. 1. Extract the square root of to within less than. SOLUTION. 3 44 429 77 10 than 1, is 6; hence is the required root. = 2. Extract the square root of 3. Find the square root of 4. Find the square root of to within less than 100. to within less than .001. to within less than . 5. Find the square root of § to within less than .0001. NOTE.-The approximate root of a fraction is usually found by reducing the fraction to a decimal, and then extracting the root by the ordinary method. CUBE ROOT. 627. The Cube Root of a number is one of the three equal factors into which the number can be resolved. 628. Any number which is not the product of three equal factors is an imperfect cube, and its cube root can be found only by approximation. 629. The reasons for the several steps in the operation of extracting the cube root, will appear by cubing a number regarded as binomial. Let 35 be written 30+ 5. Then, by cubing this number, we have By examining the composition of this number, we see that 30 27000 30 × 30 × 3 × 5 = 13500 Cube of the tens. Three times the square of the tens multiplied by the units. 30 × 3 × 52 = 2250 Three times the tens multiplied by Or, (30 + 5)3 303 + (302 × 3) × 5 + (30 × 3) × 52 + 53. Hence the Binomial Formula: The Cube of a Binomial is equal to the cube of the first term, plus three times the square of the first term multiplied by the second, plus three times the first term multiplied by the square of the second, plus the cube of the second. And, whatever the number of terms, we have the general formula: The Cube of any Polynomial is equal to the cube of the first term, plus three times the square of the first term into the second, plus three times the first into the square of the second, plus the cube of the second; plus three times the square of the SUM OF THE FIRST TWO into the third, plus three times the SUM OF THE FIRST TWO into the square of the third, plus the cube of the third, etc. Now, if from 30 + (302 × 3) × 5 + (30 × 3) × 5+ 53, we subtract 303, or the cube of the first term, there will remain (302 × 3) × 5 + (30 × 3) × 52 + 5', in which we observe that 5 is a factor of every part. The remainder may, therefore, be written thus: (30a × 3 + 30 × 3 × 5 + 52) × 5. Hence, the remainder is the product of two factors, one of which is composed of three times the square of the tens, plus three times the tens multiplied by the units, plus the square of the units; and the other is the units. Every number may be regarded as a binomial. Thus, 325 is 32 tens and 5 units. But, the development of this number, as a trinomial (300 + 30 + 5), will give— 3003 + 3 (3002 × 20) + 300 × 3 × 202 + 203 + 3 (300 + 20)2 × 5 + 3 (300 +20) × 52 + 53, which contains the cube of the binomial, 300 + 20 = 3003 + 3 (3002 × 20) + 300 × 3 × 202 + 203, and the remainder 3 (300 + 20)2 × 5 + 3 (300 + 20) × 52 + 53, or, Therefore, the remainder is the product of two factors, one of which is three times the square of the first two terms, plus three times the first tino multiplied by the third, plus the square of the third; and the other is the third term; and so of any number whatever. Hence the formula applies in all cases. 630. The involution of a binomial may be illustrated geometrically as follows: Let figure A represent a cube whose edge is 30 inches. Then, its solid contents are 303 27000 cubic inches. = Now, suppose this cube to be increased to a cube of 35 inches, by additions upon three of its sides. These additions may be represented, 1st. By the three parallelopipedons, figures B, C, D, each 30 inches square and 5 inches thick. Their solid contents are 302 x 5 x 3 = 13500 cubic inches. 2d. By the three parallelopipedons, E, F, G, each 30 inches long by 5 inches square. Their solid contents are 30 × 5a × 3 = 2250 cubic inches. 3d. By the small cube, figure H, whose edge is 5 inches, and solid contents 5 = 125 cubic inches. The cube thus constructed is represented by figure I, a cube of 35 inches = 42875 cubic inches. 631. By reversing this process of involution, we extract the cube root of any number. First, as in square root, it is necessary to distinguish the parts of the power from which the several parts of the root are derived. This is done by separating the given number into periods of three figures each, the reason for which is explained as follows: The cube of any number expressed by one figure will always be found in the first three places of the power; the cube of any number expressed by two figures, in the first six places; and so on, every three places in the power giving one figure in the root. This may be easily demonstrated by cubing the lowest and highest numbers consisting of one, two, three, or more figures; thus Roots, 1 9 10 99 100 999 Cubes, 1 729 1000 970299 1000000 997002999 The left-hand period will be a partial period consisting of only one, or two figures, when its root is less than 5. 632. From these principles we derive the following RULE.-I. Separate the given number into periods of three figures each, by placing a dot over units, another over thousands, |