EXTRACTION OF THE SQUARE ROOT. EXTRACTING the Square Root is to find out such a number as being multiplied into itself, the product will be equal to the given number. Rule. First, Point the given number, beginning at the unit's place, then to the hundreds, and so upon every second figure throughout. Secondly, Seek the greatest square number in the first point towards the left hand, placing the square number under the first point, and the root thereof in the quotient; subtract the square umber from the first point, and to the remainder bring down the next point, and call that the resolvend. Thirdly, Double the quotient, and place it for a divisor on the left hand of the resolvend; seek how often the divisor is contained in the resolvend (preserving always the unit's place,) and put the answer in the quotient, and also on the right hand side of the divisor; then multiply by the figure last put in the quotient, and subtract the product from the resolvend; bring down the next point to the remainder (if there be any more) and proceed as before. ROOTS. SQUARES. 1. 2. 1. 4. 3. 4. 5. t. 7. 8. 9. 9. 16. 25. 36. 49. 64. 81. EXAMPLES. Ans. 327. Ans. 1506,23+ Ans. 2756,228+. Ans. 6031. Ans. 4698. 2. What is the square root of 106929 ? 3 What is the square root of 2268741? 4. What is the square root of 7596796? 5. What is the square root of 36372961? 6. What is the square root of 22071204? When the given number consists of a whole number, and decimals together, make the number of decimals even by adding cyphers to them; so that there may be a point fall on the unit's place of the whole number. 7. What is the square root of 3271,4007? 8. What is the square root of 4795,25731! 9. What is the square root of 4,372594 ? Ans. 57,19+. Ans. 69,847+. Ans. 2,091+. 10. What is the square root of 2,2710957? 11. What is the square root of ,00032754? 12. What is the square root of 1,270054 ? Ans. 1,50701+. Ans. ,01809+. To extract the Square Root of a VULGAR FRACTION. RULE. Reduce the fraction to its lowest terms, then extract the square root of the numerator for a new numerator, and the square root of the denominator, for a new denominator. If the fraction be a surd (i. e.) a number where a root can never be exactly found, reduce it to a decimal, and extract the root from it. To extract the Square Root of a MIXED NUMBER. RULE 1. Reduce the fractional part of the mixed number to its lowest terin, and then the mixed number to an improper fraction. 2. Extract the root of the numerator and denominator for a new numerator and denominator. If the mixed number given be a surd, reduce the fractional part to a decimal, annex it to the whole number, and extract the square root therefrom. EXAMPLES. 19. What is the square root of 512 20. What is the square root of 27? 21. What is the square root of 943 SURDS. 22. What is the square root of 8511? 23. What is the square root of 8 24. What is the square root of 63? Ans. 2,9519+. Ans. 2,5819+. To find a mean proportional between any two given numbers. RULE. The square root of the product of the given numbers is the mean proportional sought. EXAMPLES. 27. What is the mean proportional between 3 and 12? Ans. 3×12 36 then 366 the mean proportional. 28. What is the mean proportional between 4276 and 849? Ans. 1897,4+. To find the side of a square equal in area to any given superficies. RULE. The square root of the content of any given superficies, is the square equal sought EXAMPLES. 29. If the content of a given circle be 160, what is the side of the square? Ans. 12,61911. 30. If the area of a circle is 750, what is the side of the square equal? Ans. 27,38612. The Area of a Circle given to find the Diameter. RULE. AS 355: 452, or, as 1: 1,273239 :: so is the area : to the square of the diameter:-or, multiply the square root of the area, by 1,12837, and the product will be the diameter. EXAMPLES. 31. What length of cord will be fit to tie to a cow's tail, the other end fixed in the ground, to let her have liberty of eating an acre of grass and no more, supposing the cow and tail to be 5 yards? Ans. 6,136 perches. The Area of a Circle given to find the Periphery or Circumference. RULE. AS 113: 1420, or, as 1: 12,56637:: the area to the square of the periphery,―or, multiply the square root of the area by 3,5449, and the product is the circumference. EXAMPLES. 32. When the area is 12, what is the circumference? Ans. 12,2798. 33. When the area is 160, what is the periphery? Ans. 44839. Any two sides of a right angled triangle given to find the third side. 1. The Base and Perpendicular given to find the Hypothenuse. RULE. The square root of the sum of the squares of the base and perpendicular is the length of the hypothenuse. EXAMPLES. 34. The top of a castle from the ground is 45 yards high, and surrounded M h a ditch do yards broad; what length must a ladder be to each from the outside of the ditch to the top of the case. Ans. 75 yards. |