5. 138384. 6. 390625. Ans. 85. 8. 5764801. Ans. 2401. Ans. 99. 9. 43046721. Ans. 6561. Ans. 217. 10. 49042009. Ans. 7003. Ans. 372. 11. 1061326084. Ans. 32578. Ans. 30709. EXTRACTION OF THE SQUARE ROOT OF FRACTIONS. 4 4 ART. 177. Since X5, therefore, the square root of 25 Hence, When both terms of a fraction are perfect squares, its square root will be found by extracting the square root of both terms. Before attempting to extract the square root of a fraction, it should be reduced to its lowest terms, unless both numerator and denominator are perfect squares. The reason for this will be seen by the following example. Find the square root of 20. Here 20-4X5 45 9X5 Now, neither 20 nor 45 are perfect squares; but, by canceling the common factor 5, the fraction becomes, of which the square root is 3. When both terms are perfect squares, and contain a common factor, the reduction may be made either before, or after the square root is extracted. ART. 178. A number whose square root can be ascertained exactly, is termed a perfect square. Thus, 4, 9, 16, &c., are perfect squares. Such numbers are comparatively few. A number whose square root cannot be ascertained exactly, is termed an imperfect square. Thus, 2, 3, 5, 6, &c., are imperfect squares. Since the difference of two consecutive square numbers, a2 and a2+2a+1, is 2a+1; therefore, there are always 2a imperfect squares between them. Thus, between the square of 5 (25) and the square of 6 (36), there are 10 (2a=2×5) imperfect squares. A root which cannot be expressed exactly, is called a radical, or surd, or irrational root. The root obtained is also called an approximate value, or approximate root. Thus, √2 is an irrational root; it is 1.414+. The sign is sometimes placed after an approximate root, to denote that it is less than the true root; and the sign, that it is greater than the true root. ART. 179. To prove that the square root of an imperfect square cannot be a fraction. REMARK.It might be supposed, that when the square root of a whole number cannot be expressed by a whole number, that it might be found exactly equal to some fraction. That it cannot, will now be shown. Let c be an imperfect square, such as 2, and, if possible, let its square root be equal to a fraction which is supposed to be in its lowest terms. a Then C and c a2 by squaring both sides. Now, by supposition, a and b have no common factor, therefore their squares, a2 and b2, can have no common factor, since to square a number, we merely repeat its factors. Consequently, a2 must be in its lowest terms, and cannot be equal to a whole b2 number. Therefore, the equation c is not true; and hence 62 the supposition on which it is founded is false, that is, the suppo α sition that is not true; therefore, the square root of an b imperfect square cannot be a fraction. APPROXIMATE SQUARE ROOTS. ART. 180. To explain the method of finding the approximate square root of an imperfect square, let it be required to find the square root of 5 to within }. If we reduce 5 to a fraction whose denominator is 9 (the square of 3, the denominator of the fraction ), we have 5-45. Now the square root of 45 is greater than 6 and less than 7; therefore the square root of 45 is greater than §, and less than}; hence §, or 2, is the square root of 5 to within . To generalize this explanation, let it be required to extract the square root of a to within a fraction 1 n an2 We may write a (Art. 127) under the form and if we de an2 n2 note the entire part of the square root of an2 by r, the number an will be comprised between r2 and (r+1)2; therefore and (+1); hence the square will be comprised between n2 an2 n2 root of will be comprised between 2 and n2 But the difference between n r+1 n resents the square root of a to within is therefore rep n n From this we derive the following RULE FOR EXTRACTING THE SQUARE ROOT OF A WHOLE NUMBER TO WITHIN A GIVEN FRACTION. Multiply the given number by the square of the denominator of the fraction which determines the degree of approximation; extract the square root of this product to the nearest unit, and divide the result by the denominator of the fraction. 1. Find the square root of 3 to within. 2. Find the square root of 10 to within 1. 3. Find the square root of 19 to within. 4. Find the square root of 30 to within 5. Find the square root of 75 to within Since the square of 10 is 100, the square of 100, 10000, and so on, the number of ciphers in the square of the denominator of a decimal fraction, is equal to twice the number in the denominator itself. Therefore, when the fraction which determines the degree of approximation is a decimal, it is merely necessary to add two ciphers for each decimal place required; and, after extracting the square root, to point off from the right one place of decimals for each two ciphers added. 6. Find the square root of 3 to five places of decimals. Ans. 1.73205. 7. Find the square root of 7 to five places of decimals. 3.- Find the square root of 50. 9. Find the square root of 500. Ans. 2.64575. Ans. 7.071067+. Ans. 22. 360679+. ART. 181. To find the approximate square root of a fraction. 1. Let it be required to find the square root of to within 4=4×7=28. Now, since the square root of 28 is greater than 5 and less than 6, the square root of 28 is greater than and less than ; therefore is the square root of to within less than . From this it is evident, that if we multiply the numerator of a fraction by its denominator, then extract the square root of the product to the nearest unit, and divide the result by the denominator, the quotient will be the square root of the fraction to within one of its equal parts. 30. 10' It is obvious that any decimal may be written in the form of a common fraction, and having its denominator a perfect square, by adding ciphers to both terms. Thus .3=1% 100; .156=10000, and so on. Therefore, the square root of a decimal may be found, as in the method of finding the approximate square root of a whole number (Art. 180), by annexing ciphers to the given decimal, until the number of decimal places shall be equal to double the number required in the root. Then, after extracting the root, pointing off from the right the required number of decimal places. 5. Find the square root of .4 to six places. 6. Find the square root of .35 to six places. Ans. .632455+. The square root of a whole number and a decimal may be found in the same manner. Thus, the square root of 1.2 is the same as the square root of 1.20=138, which, extracted to five places, is 1.09544+. 7. Find the square root of 7.532 to five places. Ans. 2.74444+. When the denominator of a fraction is a perfect square, its square root may be found by extracting the square root of the numerator to as many places of decimals as are required, and dividing the result by the square root of the denominator. Or, by reducing the fraction to a decimal, and then extracting its square root. When the denominator of the fraction is not a perfect square, the latter method should be used. 5 5 2.23606+== - 8. Find the square root of to five places. 16 4 .3125=.55901+. =.55901+. Ans. .774596-+. Ans. 1.11803+. Ans. 1.903943+. Ans. 3.349958+. Ans. 0.645497+. Ans. 4.168333+. EXTRACTION OF THE SQUARE ROOT OF ALGEBRAIC QUANTITIES. EXTRACTION OF THE SQUARE ROOT OF MONOMIALS. ART. 182. From Art. 172, it is evident that to square a monomial, we must square its coefficient, and multiply the exponent of each letter by 2. Thus, (3mn2)2-3mn2X3mn2=9m2n^. Ex Therefore 9m2n1=3mn2. Hence, we have the following RULE FOR EXTRACTING THE SQUARE ROOT OF A MONOMIAL. tract the square root of the coëfficient and divide the exponent of each letter by 2. Since +ax+a+a2, -ax-a=+a2; therefore, a2=+a, or —a. Hence the square root of any positive quantity is either plus, or minus. This is generally expressed by writing the double sign before the square root. Thus, √4a2+2a; which is read plus or minus 2a. If a monomial is negative, the extraction of the square root is impossible, since the square of any quantity, either positive or negative, is necessarily positive. Thus √4, a2b2, √b, are algebraic symbols, which indicate impossible operations. Such expressions are termed imaginary quantities. An example of their occurrence always arises, in proceeding to find the value of the unknown quantity in an equation of the second degree, when some absurdity, or impossibility exists in the equation, or in the problem from which it was derived. |