« ΠροηγούμενηΣυνέχεια »
64 contains 16 four times, and the product of 64 multiplied by 4, must contain 16 sixteen times, but the product of 16 multiplied by 16, is made up of 16 sixteen times.
It is frequently more convenient to find the powers of numbers by the method pursued in the 2d operation. -
20. What is the 6th power of 279 7 - * Ans. 4712655.7843°7'34'321. 21. Required the difference between the 2d and 4th powers of 27. - Ans. 530°712. 22. What is the quotient of the 4th power of 12, divided by the cube of 42 Ans. 324. 23. 11°-i-11° equals what number 2 * - Ans. 11. 24. What is the square of 25,62 Ans. 655,36. 25. What is the cube of 7 ? Ans. 343. 26. What is the square of 14; ? Ans. 210,25.
JNote.—When the exact value of a vulgar fraction can be found in a decimal, the better way will generally be, to change it before finding the power.
27. What is the cube of 2; ? Ans. 17,576. 28. What is the cube of ,001 ? Ans, ,000000001. 29. How many square rods in a garden in the form of a square, if the length of one side be 7 rods? Ans. 49.
Let figure A represent the garden mentioned in the last question. The garden being in the form of a square, the length is 7 rods. A section on one side, one rod in width, contains 7 square rods, and in the whole garden there are 7 such sections, consequently multiplying 7 by 7, the product will be the number of square rods contained in the whole garden. Since the length of one side of a square multiplied by itself, produces the whole contents of that square, the length of one side of any square is a root, and the qontents of that square the 2d power of the same root. It is from this circumstance that the 2d power of a root is called a square. It may further be observed, that the root consists of dimensions in long measure, but the 2d power is in the dimensions of square measure. It has already been shown, that the product of a proper fraction multiplied by a proper fraction, is less than the given multiplicand, or multiplier, consequently, the square or any power of a proper fraction, must be a less part of an unit than itself. Let figure B represent a square surface of any kind, as a piece of board or IB paper, having gb, b.f., f h, k G- al b g, each one foot in length, the whole surface would con* d tain one square foot. If the side of a square be # of a t S foot in length, we still find the area by multiplying the © length of one side by itself. £ The square of , is . If b s be half of bf, it is 4 of a - foot, and multiplying the length of b s into itself, gives the area of the small square d, which is 3 part of the whole square B. It may be proper to observe, that the square of ; of a foot in long measure is 4 of a square foot, since d is # part of B. . Next let us suppose c f to be # of a foot in length, and o a square, it can be only to part
of the whole square B, for ; x +=To the square of +. When the numerator of a fraction has more than one fractional unit in it, the square of it will be represented by a number of small squares equal to the number of fractional units in the 2d power of the given fraction, but all these small squares will still be less than a square of an unit in the same measure. It will be useful for the learner to square a number of fractions whose numerators have more than one fractional unit, and draw figures corresponding to them.
30. How much land in a square field whose side is 134 rods 2 - Ans. 1 124°F acres. 31. How much land in a yard in the form of a square, if the side is or of a mile ! Ans. rior of a square mile, or or of an acre, or 67 or s, rods.
4 cubick body, or cube, is a body with 6 equal sides, or faces, the opposite sides being parallel to each other.
32. What is the number of cubical or solid feet in a cubical block, the length of whose side is 3 feet 7 Let the figure C represent C - l a cubical i. 3 ft. . 3 ft. lc [.. wide, and 3 ft. thick. It was 1 proved (Illustration page - 97,) that the number of solid feet in 1 foot of the thickness on one side of a solid body, is P equal to the number of square feet on that side ; therefore if we square 3, the square will represent the number of solid feet in one foot of the thickness of the supposed block C; and since the thickness of a cube is equal to the length, there will be 3 sections of the block, each containing 9 solid feet; consequently multiplying 9 the square of 3 by 3, will give the number ... feet in the whole block which is 27. Or in other words, the cube of the length of one side
of a cubical body, expresses the whole centents of that body. We supposed lp to be 3 ft. in length, which is 1 yard. Let li be 1 foot in length, and it is # of a yard. Now if we wish to find what the cube of , of the length of the side is, we must cube , which is or of a solid yard, or 1 solid foot, since 27 solid feet make 1 solid yard. The cube of 4 would be a block each. side of which would be equal to k. Raising the length of one side of a cubical body to the third power, will always give the solid contents of that body, for the same reasons which have just been explained. This explains the reasons why the third power of a number is called the cube of that number. 33. How many cubical feet in a cubical vessel, whose side is 14 feet in length 2 Ans. 2744. 34. How many solid feet in a cubical block, the side of which measures 14 inches in length ? Ans. 1 solid foot 1016 solid inches, or 1444 cubical feet.
Involution is the method of finding a required power from a given root; evolution is finding a root of a given power; in one we involve or mix numbers, in the other unfold them. -
!f a power be divided by the root, and that quotient by the root, and so on, until the number of divisions should equal the number of times the root was multiplied into itself to create the power, the last quotient would be the root. But as the root is not supposed to be known, a method has been invented for finding it. Were the number known which produced the power, the root would be known, which would render an operation useless.
The square of any single figure cannot exceed two places of figures, since the square of 9 is only 81 ; the square root therefore of any three places of figures cannot be expressed by one figure. The square of any number can have no more than twice as many places of figures as the root, for the product of the left hand figure in the multiplier can never extend farther to the left of the multiplicand, than the number of places in the multiplicand. From these facts it appears, that the number of places in the root of the 2d power, when the 2d power has an even number of places, can never exceed half the number of places in the power; and when the number of places in the power is odd, the number of places in the root must be one more than half the greatest even number of places.
1. Place a point overevery other figure beginning with units’ place, whether the number be a mixed or whole number. 2. Find the root of the greatest square in the left hand period; subtract the square of that root from the period, and to the remainder annex the next period. 3. Double the root already found for a divisor, and ascertain how many times the divisor can be taken from the dividend, (formed by annexing the next period,) omitting the right hand figure ; place the figure representing the number of times for the next figure of the root; also annex the same figure to the divisor. Multiply the divisor by the last figure placed in the root, and subtract the product from the dividend. To the remainder bring down the next period, if any, and proceed as before.
JNote—The most convenient method of doubling the root for a divisor after the first has been found, is to double the right hand figure of the last, which may be continued for a divisor by making this increase,