Extraction of the square root of Fractions. 119. Since the square or second power of a fraction is obtained by squaring the numerator and denominator separately, it follows that the square root of a fraction will be equal to the square root. of the numerator divided by the square root of the denominator. We can But if neither the numerator nor the denominator is a perfect square, the root of the fraction cannot be exactly found. however, easily find the exact root to within less than one of the equal parts of the fraction. To effect this, multiply both terms of the fraction by the denominator, which makes the denominator a perfect square without altering the value of the fraction. Then extract the square root of the perfect square nearest the value of the numerator, and place the root of the denominator under it; this fraction will be the approximate root. Thus, if it be required to extract the square root of 3 we mul. 5 15 tiply both terms by 5, which gives : the square nearest 15 is 25 4 16: hence is the required root, and is exact to within less 5 120. We may, by a similar method, determine approximatively the roots of whole numbers which are not perfect squares. Let it be required, for example, to determine the square root of an entire number a, nearer than the fraction that is to say, to find a 1 n number which shall differ from the exact root of a, by a quantity entire part of the root of an2, the number an2 will then be compris Multiply the given number by the square of the denominator of the fraction which determines the degree of approximation: then extract the square root of the product to the nearest unit, and divide this root by the denominator of the fraction. Suppose, for example, it were required to extract the square root of 59, to within less than 1 12 Let us repeat on this example, the demonstration which has just been made. 121. The manner of determining the approximate root in decimals, is a consequence of the preceding rule. To obtain the square root of an entire number within 10' 100 1 &c.—it is necessary according to the preceding rule to mul1000' tiply the proposed number by (10)3, (100)2, (1000)3 .. or, which is the same thing, add to the right of the number, two, four, six, &c. ciphers: then extract the root of the product to the nearest unit, and divide this root by 10, 100, 1000, &c., which is effected by pointing off one, two, three, &c., decimal places from the right hand. . Example 1. To extract the square root of 7 to within Having added four ciphers to the right hand of 7, it becomes 70000, whose root extracted to the nearest unit is 264, which being divided by 100 gives 2,64 for the answer, which 1 100 7.00.00 2,64 4 46 300 276 524 2400 2096 304 Rem REMARK. The number of ciphers to be annexed to the whole number, is always double the number of decimal places required to be found in the root. 122. The manner of extracting the square root of decimal frac. tions is deduced immediately from the preceding article. Let us take for example the number 3,425. This fraction is equivalent to 3425 Now 1000 is not a perfect square, but the denominator may be made such without altering the value of the fraction, by multiplying both the terms by 10; this gives 34250 10000 If greater exactness be required, it will be necessary to add to the number 3,4250 so many ciphers as shall make the periods of decimals equal to the number of decimal places to be found in the root. Hence, to extract the square root of a decimal fraction: Annex ciphers to the proposed number until the decimal places shall be even, and equal to double the number of places required in the root. Then extract the root to the nearest unit, and point off from the right hand the required number of decimal places Ex. 1. Find the 3271,4707 to within 01. Ans. 57,19. 2. Find the √31,027 to within ,01. 3. Find the √0,01001 to within ,00001. Ans. 5,57. Ans. 0,10004. 123. Finally, if it be required to find the square root of a vulgai fraction in terms of decimals: Change the vulgar fraction into a decimal and continue the division until the number of decimal places is double the number of places required in the root. root of the decimal by the last rule. Then extract the Ex. 1. Extract the square root of 11 14 to within ,001. This number, reduced to decimals, is 0,785714 to within 0,000001; but the root of 0,785714 to the nearest unit, is ,886 : hence 0,886 is Extraction of the Square Root of Algebraic Quantities. 124. We will first consider the case of a monomial; and in order to discover the process, see how the square of the monomial is formed. By the rule for the multiplication of monomials (Art. 41.), we have (5a2b3c)=5a2bc5a2b3c=25a1b°c2; that is, in order to square a monomial, it is necessary to square its co-efficient, and double each of the exponents of the different letters. Hence, to find the root of the square of a monomial, it is necessary, 1st. To extract the square root of the co-efficient. 2d. To take the half of each of the exponents. Thus, √64ab8a3b2; for 8a3b28a3b3-64a3b1. |