The smallest sum on deposit during the first interest term was $50. The interest on $50 for 3 mo. at 4% is $0.50, which, added to the balance on deposit, makes $70.50.

The smallest sum on deposit during the second interest term was $40.50. The interest on $40.50 for 3 mo. at 4% is $0.40, which, added to the balance on deposit, makes $75.90.

The smallest sum on deposit during the third interest term was $75.90. The interest on $75.90 for 3 mo. at 4% is $0.76, which, added to the balance on deposit Oct. 1, 1897, makes $76.66.

EXERCISE 138.

Find the balance on deposit Jan. 1, 1898, on the following account :

1. Interest being 4%, computed quarterly. Deposited Jan. 1, 1897, $125; Mar. 22, 1897, $40; June 8, 1897, $35; July 30, 1897, $85; Sept. 24, 1897, $65. Withdrawn Apr. 2, 1897, $110; June 30, 1897, $40; Oct. 22, 1897, $10; Dec. 17, 1897, $25.

2. Interest being 3%, computed quarterly. Deposited Jan. 1, 1897, $200; Feb. 14, 1897, $125; Mar. 10, 1897, $75; May 31, 1897, $50; Aug. 2, 1897, $100. Withdrawn May 7, 1897, $25; June 22, 1897, $40; Oct. 2, 1897, $50; Nov. 4, 1897, $65; Dec. 14, 1897, $75.

3. Interest being 3 %, computed semi-annually. Deposited Jan. 1, 1897, $425; May 10, 1897, $15; Sept. 24, 1897, $200; Oct. 5, 1897, $25; Nov. 15, 1897, $65. Withdrawn Feb. 1, 1897, $25; Mar. 20, 1897, $45; Aug. 2, 1897, $50; Aug. 28, 1897, $125; Dec. 10, 1897, $100.

4. Interest being 3%, computed annually. Deposited Jan. 1, 1897, $266.50; May 3, 1897, $122.50; Aug. 2, 1897, $57; Aug. 9, 1897, $108; Sept. 4, 1897, $64.50. Withdrawn June 15, 1897, $40; Oct. 8, 1897, $75; Nov. 1, 1897, $60; Dec. 4, 1897, $85; Dec. 20, 1897, $142.

CHAPTER XV.

POWERS AND ROOTS.

532. The square of a number is the product of two factors, each equal to the number (§ 69).

are

Thus, the squares of 1, 2, 3, 4, 5, 6, 7, 8, 9, 10,

1, 4, 9, 16, 25, 36, 49, 64, 81, 100.

533. The square root of a number is one of the two equal factors of the number (§ 69).

are

Thus, the square roots of 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10.

534. The square root of a number is indicated by the radical sign, or by the fractional exponent.

Thus, √27, or 271, means the square root of 27.

535. Since 35 = 305, the square of 35 may be

536. Since every number of two or more figures may be regarded as composed of tens and units, if we represent the number of tens by a and the number of units by b,

(a + b)2 = a2 + 2 ab+b2. Hence,

The square of a number is equal to the square of the tens, plus twice the tens multiplied by the units, plus the square of

the units.

537. The first step in extracting the square root of a number is to separate the figures of the number into groups.

Since 112, 100 = 102, 10,000 = 1002, and so on, it is evident that the square root of any number between 1 and 100 lies between 1 and 10; of any number between 100 and 10,000 lies between 10 and 100. In other words, the square root of any integral number expressed by one or two figures is a number of one figure; expressed by three or four figures is a number of two figures, and so on.

If, therefore, an integral number is divided into groups of two figures each, from the right to the left, the number of figures in the root will be equal to the number of groups of figures. The last group to the left may have one or two figures.

Example. Find the square root of 1225.

SOLUTION. The first group, 12, contains the square of the tens' number of the root.

The greatest square in 12 is 9, and the square root of 9 is 3. Hence, 3 is the tens' figure of the root.

12 25(35 9 65)3 25 325

The square of the tens is subtracted, and the remainder contains twice the tens X the units + the square of the units. Twice the 3 tens is 6 tens, and 6 tens is contained in the 32 tens of the remainder 5 times. Hence, 5 is the units' figure of the root. Since twice the tens × the units + the square of the units is equal to (twice the tens + the units) × the units, the five units are annexed to the 6 tens, and the result, 65, is multiplied by 5.

538. The same method will apply to numbers of more than two groups of figures, by considering the part of the root already found as so many tens with respect to the next figure of the root.

Example. Extract the square root of 7,890,481.

7 89 04 81 (2809

4

48)3 89

3 84 5609)5 04 81

5 04 81

SOLUTION. When the third group, 04, is brought down, and the divisor, 56, formed, the next figure of the root is 0, because 56 is not contained in 50. Therefore, 0 is placed both in the root and the divisor, and the next group, 81, is brought down.

539. If the square root of a number has decimal places, the number itself will have twice as many.

Thus, if 0.11 is the square root of some number, the number will be (0.11)2 = 0.11 × 0.11 = 0.0121. Hence, if a given number contains a decimal, we divide it into groups of two figures each, beginning at the decimal point and marking toward the left for the integral number, and toward the right for the decimal. The last group of the decimal must have two figures, a cipher being annexed if necessary. Example. Extract the square root of 52.2729.

52.27 29(7.23

49

142) 3 27

284

1443) 43 29 43 29

SOLUTION. It will be seen from the groups of figures that the root will have one integral and two decimal places.

540. If a number is not a perfect square, ciphers may be annexed, and an approximate value of the root found. Example. Extract the square root of 17 to six places. 17.00 00 00 (4.123106

SOLUTION. In this example, after finding four figures of the root, the other three are found by common division.

The rule in such cases is that one less than the number of figures already obtained may be found without error by division, the divisor to be employed being twice the part of the root already found.

541. The square root of a common fraction is found by extracting the square root of the numerator and of the denominator. If the denominator is not a perfect square, multiply both terms of the fraction by a number that will make the denominator a perfect square, or reduce the fraction to a decimal and extract the root of the decimal.

542. RULE FOR SQUARE ROOT. Separate the number into groups of two figures each, beginning at the units.

Find the greatest square in the left-hand group and write its root for the first figure of the required root.

Square this root, subtract the result from the left-hand group, and to the remainder annex the next group for a dividend.

For a partial divisor, double the root already found, considered as tens, and divide the dividend by it. The quotient (or the quotient diminished) will be the next figure of the

root.

To this partial divisor add the last figure of the root for a complete divisor. Multiply this complete divisor by the last figure of the root, subtract the product from the dividend, and to the remainder annex the next group for a new dividend.

Proceed in this manner until all the groups have been thus annexed. The result will be the square root required.

NOTE 1. If the number is not a perfect square, annex groups of zeros and continue the process.

NOTE 2. If the given number contains a decimal, divide it into groups of two figures each, beginning at the decimal point and marking toward the left for the integral number and toward the right for the decimal number. The last group on the right of the decimal must contain two figures, a zero being annexed if necessary.