hence, the nth power of the mth power of a number is equal to the mnth power of this number. Let us see if the converse of this is also true. then raising both members to the nth power, we have, from the definition of the nth root, m√ a = bn ; and by raising both members of the last equation to the mth power a = bmn Extracting the mnth root of both members of the last equation, And since each is equal to b. Therefore, the nth root of the math root of any number, is equal to the mnth root of that number. in a similar manner, it might be proved that REMARK. Although the successive roots may be extracted in any order whatever, it is better to extract the roots of the lowest degree first, for then the extraction of the roots of the higher degrees, which is a more complicated operation, is effected upon numbers containing fewer figures than the proposed number. Extraction of Roots by Approximation. 144. When it is required to extract the nth root of a number which is not a perfect nth power, the method already explained, will give only the entire part of the root, or the root to within less than 1. As to the part which is to be added, in order to com plete the root, it cannot be obtained exactly, but we can approximate to it as near as we please. Let it be required to extract the nth root of a whole number, denoted by a, to within less than a fraction ; that is, so near, 1 p If we denote by r the root of the greatest perfect nth power in ap", the number ( +1)" pn ах 20 = a, will be comprehended between and pn ; therefore, the a will be comprised between the two numbers 1 p will be greater than the difference between and the true γ Ρ root. Hence, is the required root to within less than the Ρ To extract the nth root of a whole number to within less than 1 a fraction multiply the number by p"; extract the nth root of the product to within less than 1, and divide the result by p. Extraction of the nth Root of Fractions. 145. Since the nth power of a fraction is formed by raising both terms of the fraction to the nth power, we can evidently find the nth root of a fraction by extracting the nth root of both terms. If both terms are not perfect nth powers, the exact nth root cannot be found, but we may find its approximate root to within less than the fractional unit, as follows: α Let represent the given fraction. If we multiply both b Let r denote the nth root of the greatest nth power in a¿”~1; and consequently, will be the nth root of b Multiply the numerator by the (n-1)th power of the denomi nator and extract the nth root of the product: Divide this root by the denominator of the given fraction, and the quotient will be the approximate root. When a greater degree of exactness is required than that extract the nth root of abn-1 to withir ny 1 is the root of the numerator to within less than Ρ that is the true root of the fraction to within less that bp bp EXAMPLES. 1. Suppose it were required to extract the cube root of 15, 1 to within less than We have 12' 15 × 123 15 × 1728 = 25920. Now, the cube root of 25920, to within less thar. 1 is 9; hence, the required root is, 1 2. Extract the cube root of 47, to within less than 20 We have, 47 × 203 = 47 × 8000 = 376000. Now, the cube root of 376000, to within less than 1, is 72; 3/25 25, to within less than .001. by the cube of 1000, or 1000000000, Now, the cube root of this number, = 2.920 to within less than .001. Hence, to extract the cube root of a whole number to within less than a given decimal fraction, we have the following RULE. Annex three times as many ciphers to the number, as there are decimal places in the required root; extract the cube root of the number thus formed to within less than 1, and point off from the right of this root the required number of decimal places. 146. We will now explain the method of extracting the cube root of a decimal fraction. Suppose it is required to extract the cube root of 3.1415. Since the denominator, 10000, of this fraction, is not a per fect cube, make it one, by multiplying it by 100; this is equiva lent to annexing two ciphers to the proposed decimal, which then becomes, 3.141500. Extract the cube root of 3141500, that is, of the number considered independent of the decimal point to within less than 1; this gives 146. Then dividing by 100, or 3/1000000, and we find, 3/3.1415 1.46 to within less than 0.01. Hence, to extract the cube root of a decimal fraction, we have the following RULE. Annex ciphers till the whole number of decimal places is equal to three times the number of required decimal places in the root. Then extract the root as in whole numbers, and point off the required number of decimal places. To extract the cube root of a vulgar fraction to within less than a given decimal fraction, the most simple method is, To reduce the proposed fraction to a decimal fraction, continuing the division until the number of decimal places is equal to three times the number required in the root. The question is then reduced to extracting the cube root of a decimal fraction. Suppose it is required to find the sixth root of 23, to within less than 0.01. Applying the rule of Art. 144 to this example, we multiply 23 by (100), or annex twelve ciphers to 23; then extract the sixth root of the number thus formed to within less than 1, and divide this root by 100, or point off two decimal places on the right: we thus find, 6 23 = 1.68, to within less than 0.01. EXAMPLES. 1. Find the 3/473 to within less than 2. Find the 3 3/79 to within less than .0001. Ans. 73. Ans. 4.2908. 3. Find the 6/13 to within less than .01. Ans. 1.53. Ans. 1.4429. 5. Find the 3/0.00101 to within less than .01. 6. Find the 3 to within less than .001. Ans. 0.10. Ans. 0.824. |