17328=3aa 228=3a 173508(5218.947 Resolvend. Divis. 173508=3aa+3a (7) The next figure in the Root, viz. 3, found as before is also called e; then again Sade+3eea+eee is the other Subtrahend, or number to be subducted; thus, I What is the Cube of 6.4 ? Answer 262. 144 of 7612.812161 ? 7 What is the Cube Root of 7612181.7612? 8 What is the Cube Root of 61218.00121? 9 What is the Cube Root of 7121.1021698 ? 10 What is the Cube Root Answ. 19.67+ Answ. 196.71+ Answ. 39.41+ Answ. 19.238+ Answ. 22.89+ 12 What is the Cube Root of .0069761218? Answ. .495+ Answ. .19107+ 13 If a cubical piece of timber be 41 inches long, 41 inches broad, and 41 inches deep, how many cubical inches doth it contain? Ans. 68921 cubical inches. 14 Suppose a Cellar to be dug, that shall be 12 feet every way.in length, breadth and depth; how many solid feet of earth must be taken out to complete the same? Ans. 1728. 15 Suppose a stone of a cubic form to contain 474552 solid inches; what is the superficial content of one of its sides? Ans. 6084 inches. OF THE CUBE-ROOT OF A VULGAR FRACTION. Q. How do you extract the Cube Root of a Vulgar Fraction? A. 1 Reduce the Fraction to its lowest terms. 2 Extract the Cube Roots of the Numerator and Denominator for a new Numerator and Denominator. 3 If the Fraction be a Surd reduce it to a Decimal, and then extract the Cube Root from it. 4 The Decimal Fraction must consist of Ternaries of places; as three, six, nine, &c. EXAMPLES. 1 What is the Cube Root of Answer 2 What is the Cube Root of 1944? Ans. SURDS. 4 What is the Cube Root of ? Ans. .763+ 5 What is the Cube Root of ? Ans. 94946 What is the Cube Root of? Ans. 693+ OF THE CUBE ROOT OF A MIXED NUMBER. Q. How do you extract the Cube Root of a mixt number? A. 1 Reduce the fractional part to its lowest terms. 2 Reduce the mixt number to an improper fraction. 3 Extract the Cube Roots of the Numerator and Denominator, for a new Numerator and Denominator. 4 If the mixt number given be a Surd, reduce the fraetional part to a Decimal, and annex it to the whole number, and extract the Cube Root from the whole. EXAMPLES. 1 What is the Cube Root of 5781?Ans. 8 Q. SURDS. 2 4 What is the Cube Root of 8 Ans 2.0135 What is the Cube Root of 7? Ans. 1.966+ OF THE BIQUADRATE ROOT. WHAT is the Biquadrate Number? A. Any Number involved four times produces a Biquadrate. Q. How is the Biquadrate Root extracted? A. First extract the Square Root of the given Resolvond, and then extract the Square Root of that Square Root, for the Biquadrate Root required. EXAMPLES. 1 What is the Biquadrate of 48? Ans. 5308416. 2 What is the Biquadrate of 96? Ans.84934656. 3 What is the Biquadrate Root of 5308416? Ans. 48. 4 What is the Biquadrate Root of 84934656? Ans. 96. 5 What is the Biquadrate Root Ans. 384. A. Any Number involved 5 Times produces a Sirsolid. Q. How is the Sursolid Root, or the Root of any other higher Power extracted?.. A. By the following general Rules. If any even power be given, let the Square Root of it be extracted, which reduces it to half of the given Power, then the Square Root of that Power reduces it to half of the same power, and so on till you come to a square of a Cube. For example: Suppose a 24th Power be given the Square Root of that reduces it to a 12th Power, the Square Root of the 12th Power reduces it to a 6th Power; and the Square Root of the 6th Power to a Cube. 2 If any odd Power be given, as the 17th, fc. observe [1] From the Unity Place, both ways, point off at every such Number of figures as is the Index of the power for a Resolvend. [2] Seek in the Table of Powers, for such a power (being the same Power with the Index) as comes nearest the first Period, whether greater or less, calling its Root aceordingly more than just, or less than just. [3] Annex so many Cyphers to the Root, as there are Periods of whole Numbers in the given Resolvend. [4] Find the difference between the given Resolvend and the power coming nearest the first Period. [5] Whatever odd power is given, the next lowest odd power to that of the said root must be found with its annexed cyphers; i. e. if the ninth power be given, find the 7th power of the root and cyphers: if the 11th power be given, find the 9th, &c. [6] Multiply the next lowest odd power by the Index of the given power, and let that product be a divisor to the difference between the given resolvend and power first found, which depresses it to a square. [7] Point this square into periods of two figures each. [8] Then make the first root without its cypher a divisor, and ask how oft it may be found in the first period of the square. [9] If the divisor be less than just, you must multipyi the quotient figure by half the iudex, e if the index be 11, multiply the quotient figure by 5; if the index be 9, multiply it by 4, §.e. and add it to the divisor; but if it be more than just, you must subtract it from the divisor, having a cypher annexed or supposed to be annexed to the divisor; which sum or difference must be multiplied by the said quotient figure, and so continued to every new figure in the quotient. [10] If the first root with its cyphers be more than just, the quotient must be subtracted from it; but if it be less than just, it must be added to it; and the sum or difference will be the root required. 3. If an even power be giver, and the square root of that power be extracted, reduce it to an odd power; you must then proceed with that odd power as the foregoing rules direct. EXAMPLES. 1. What is the sursolid of 6436343 ? .6436343 32 the nearest sursolid, whose root and cypher is 20. 3236343 The cube of 20 is=8000 Lastly 20 23 the sursolid 1st divisor 26 2 to be rejected. Root required Note. This is a very expeditious way of extracting the roots of high powers, but it is not always exact, because (as Mr. Ward observes, for it was taken from him) there will be a remainder, and sometimes an Excess or Defect in the last Figure of the root, when the given resolvend or power hath a true root-as appears by the fifth example following, whose true root should not be 384.3, as it there stands, but 384. 2. What is the sursolid of 48? Ans. 254803968. 3. What is the sursolid root of 8153726976 ? Ans. 96. 4. What is the sursolid root of 254803968? Ans. 48. 5. What is the sursolid root of 2 8349416423424 P Ans. 384.3. OF THE SQUARE-CUBE ROOT. Q. WHAT is a Square Cube? A. Any Number involved six times, produces a Square Cube. 96. EXAMPLES. 1. What is the Square-Cube of 48? Ans. 12230590464. 2. What is the Square-cube root of 782757789696? Ans. 3. What is the Square-cube root of 12230590464? Ans. 48. 4. What is the Square-cube root of 3206175906594816 P Answer 384. OF THE SECOND SURSOLID-ROOT. Q. WHAT is the Second Sursolid ? A. Any number involved seven times produces a second Sursolid. EXAMPLES. 1. What is the 2d sursolid of 96? A. 75144747810816. 2. What is the second sursolid root of 75144747810816? Answer 96. 3. What is the second sursolid root of 587068342272 ? Answer 48. 4. What is the 2d sursolid root of 1231171548132409344 P Answer 38.42. OF THE SQUARE BIQUADRATE-ROOT. Q. WHAT is a Square Biquadrate ? A. Any number involved eight times, is a Biquadrate squared, or square biquadrate. |