19th

1 512 19683 262144 1953 125 10077696 40353607 134217728 387420489 Note.—1, 2, 3, 4, 5, 6, 7, 8, and 9 are Indices. Example 29 = 512.

PROBLEM. To find any distant powers of the roots in the Table without producing all the intermediate powers.

RULE. Take from the Table indices of which the sum is equal to the index of the required power. Then the continual multiplication of the powers belonging to such indices, will give the required power.

Examples.

What is the 15th power of 3?

Indices 7+ 8 15 the index of the required power.

Then 37 x 38 or 2187 × 6561 = 14348907 the 15th power of 3.

What is the 12th power of 4?
What is the 30th power of 2?

A. 16777216.

A.

EVOLUTION

Teaches how to find some root of a given number or power. When the given number is in the Table the root stands over it. Thus the cube root of 789 is 9.

TO EXTRACT THE SQUARE ROOT. RULE.-Put a dot over the units, and if the number consists of more than two figures, put a dot over the third from the units, and thus divide any whole number into periods by pointing from right to left; but in decimals point from left to right. Thus 27615, 0127.

Having found the greatest square, subtract it from the given number; and, having put the root to the right, to the remainder bring down the next period for a dividend. On the left of the dividend write the double of the root found for a divisor. Ask how often this double is contained in the dividend, rejecting its last figure, and place the answer after the root found and after the divisor. Multiply the new divisor by the last figure of the root, and subtract the product from the dividend. If the product is too much, put a less figure to the root, and after the divisor. To the remainder annex the next period for a new dividend, and find a new divisor by doubling the whole root and thus continue the work.

Examples.

Extract the square root Extract the square root of 2-0000(1-4142+ root. of 76184-3760(276-0158.

1

24 100 Dividend

4

PROOF.-Multiply the root by itself, and to the product add the remainder.

What is the square root of 5? A. 2.236068.

What is the square root of 3·1721812? A. 1-78106 &c. What is the square root of 761.801216? A. 27·6007 &c. What is the square root of ⚫0007612816? A. 02759 &c. PROBLEM. To extract the square root of a vulgar fraction.

RULE. If the numerator and denominator are square numbers, the root of each will be the terms of the fraction required. Thus the square root of 1 is . But if the terms of the given fraction are not squares, reduce it to a decimal by Prob. 1, page 69, and extract the root as already directed.

NOTE. The Vulgar Fractions of mixt numbers must be reduced to decimals.

What is the square root

of? A. 0645469, &c. | of 387? A. 0·72414, &c. of 79? A. 2.7961, &c. of 76 14? A. 8.7649, &c.

TO EXTRACT THE CUBE ROOT.

RULE. Put a point over the unit's place and over every third figure to the left; but in decimals over every

third to the right. Thus 61218-001210. Then find by the table the nearest less or an equal cube number to the left hand period, and put its root for the first figure of the root sought. 2.-Subtract this cube from the first period, and to the remainder bring down the next period for a dividend. 3.-Find a divisor by multiplying the square of the root by 300. 4.-Divide the dividend by this divisor and the quotient is the next figure of the root. 5.-Multiply the figure (or figures) of the root found by 30, and the product by the last figure, and write the product under the divisor. 6.-Under this last product, write the square of the last figure of the root found, and add the three numbers together. 7.-Multiply the sum of these three numbers by the last figure of the root, and subtract the product from the dividend; but if the product is greater than the dividend, put a less figure than the last to the root, and alter the work. 8.-To the remainder bring down the next period for a new dividend, with which proceed as with the first dividend, until all the periods are brought down. Prove the work by cubing the root and adding the remainder.

What is the cube root of 61218.00121 ?

3rd Divisor 3942×300=46570800 55017210 3rd D. 394x30x1= 11820

12=

1

1 × 46582621=46582621

8434589 Remainder.

What is the cube root of

7612-81861 ? A. 19.67, &c.

What is the cube root of 7612181-7612? A. 196·71, &c.

ARITHMETICAL PROGRESSION.

When a series of numbers increases by equal differences, they are said to be in Arithmetical Progression.

As 1, 4, 7, 10, 13 common difference is 3.
As 1, 1, 2, 1, 14 common difference is 4.

NOTE. The last term and the sum of all the terms may be found by addition, thus, of the series 1, 3, 5, 7, &c., to nine terms, the last term found by continual addition is 17, and the sum of all the terms is 81. But as this method, for a long series, would be tedious, the following two rules were invented to shorten such operations.

PROBLEM 1.-Given the first term, the number of terms and the common difference, to find the last term.

RULE.-Multiply the common difference, by a number which is one less than the number of terms; add the product to the first term, and the sum will be the last.

Examples.

Given the first term 3, the common difference 2, and the number of terms 9, to find the last term. Facit 19.

A man bought 100 yards of cloth, the first yard cost 10 pence, the second 13 pence, increasing 3 pence every yard; what was the price of the last yard? A. £1 5s. 7d.

John is indebted to James, and agrees to pay off the debt in one year, by paying 5 shillings the first week, 9 shillings the second, increasing the payment every following week by 4 shillings; what is the debt?

A. £10 9s. PROBLEM 2.-Given the first term, the number of terms and the common difference to find the sum of all the terms.

RULE. Find the last term by the last rule. Multiply the sum of the first and last by the number of terms, and half the product will be the sum of all the terms.

Examples.

Bought 19 yards of shalloon, and gave 1d. for the first yard, 3d. for the second, and 5d. for the third, in