II. Then point off the decimals into periods of two figures each, counting from the unit's place towards the right. If there are whole numbers they must be pointed off as in Case I. Then extract the root, as in Case I. NOTE. If the given number has not an exact root, there will be a remainder after all the periods have been brought down, in which case the operation may be extended by forming new periods of ciphers. To extract the square root of a vulgar fraction, or mixed number, we have this RULE. I. Reduce the vulgar fraction, or mixed number, to its simplest fractional form. II. Then extract the square root of the numerator and denominator separately, if they have exact roots; but when they have not, reduce the fraction to a decimal, and proceed as in Case II. Ans. 21. 4. What is the square root of 3 of 4 of 4 of 7? When there are many figures required in the root, we may, after obtaining one more than half the number required, find the rest by dividing the remainder by the last TRUE DIVISOR, deprived of its right-hand figure. This division should be performed according to the abridged method, as explained under Art. 42, page 56. In the preceding example, after obtaining 9 figures of the root, by the usual rule, we had for the remainder 23574559, the last true divisor was 66332494/9, when deprived of its right-hand figure. We then divide this remainder by this divisor, according to the method of abridged division of decimals, Art. 42, page 56, and obtained the remaining 8 figures of the root. 2. What is the square root of 3 to 10 decimals? Ans. 1.7320508076. 3. What is the square root of 0.00008876684 to 10 places of decimals? Ans. 0.0094216155. 4. What is the square root of 0.8867081113724 to 10 places of decimals? Ans. 0.9416517994. 5. What is the square root of 3.14159265 to 8 places of decimals? . 6. What is the square root of 2 to 9 places of decimals? 7. What is the square root of 100 to 15 places of decimals? 8. What is the square root of 0.365 to 7 places of decimals? EXAMPLES INVOLVING THE PRINCIPLES OF THE SQUARE ROOT. 78. A triangle is a figure having three sides, and consequently three angles. When one of the angles is right, like the corner of a square, the triangle is called a right-angled triangle. In this case the side opposite the right angle is called the hypotenuse. It is an established proposition of geometry, that the square of the hypotenuse is equal to the sum of the squares of the other two sides. From the above proposition, it follows that the square of the hypotenuse, diminished by the square of one of the sides, equals the square of the other side. a By means of these properties, it follows that two sides of right-angled triangle being given, the third side can be found. Examples. 1. How long must a ladder be, to reach the top of a house, 40 feet high, when the foot of it is 30 feet from the house? In this example it is obvious that the ladder forms the hypotenuse of a right-angled triangle, whose sides are 30 and 40 feet respectively. Therefore the square of the length of the ladder must equal the sum of the squares of 30 and 40. √2500=50=the length of the ladder. 2. Suppose a ladder, 100 feet long, to be placed 60 feet from the roots of a tree, how far up the tree will the top of the ladder reach? Ans. 80 feet. 3. Two persons start from the same place, and go, the one due north, 50 miles, the other due west, 80 miles. apart are they? How far Ans. 94.34 miles, nearly. 4. What is the distance through the opposite corners of a square yard? Ans. 4.24264 feet, nearly. 5. The distance between the lower ends of two equal raf |