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CASE III.

To extract the square root of a vulgar fraction, or mixed number, we have this

RULE

1. 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.

EXAMPLES.

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1. What is the square root of 318?

Ans. . 2. What is the square root of ? Ans. . 3. What is the square root of 431? Ans. 23. 4. What is the square root of of 4 of 4 of [?

Ans. . 5. What is the square root of 4; ?

Ans. 2:027 nearly. 6. What is the square root of 17 ?

Ans. 0-8044 nearly. 7. What is the square root of 13? Ans. 0.52 nearly.

EXAMPLES INVOLVING THE PRINCIPLES OF THE SQUARE ROOT.

133. 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.

By means of these properties, it follows that two sides of a right-angled triangle being given, the third side can be found.

EXAMPLES.

1. How long must a ladder be, to reach to 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.

30% = 900
402 = 1600

V2500=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. How far apart are they? 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 rafters, in the different sides of a roof, is 32 feet, and the height of the ridge above the foot of the rafters is 12 feet. What is the length of a rafter ?

Ans. 20 feet. 6. What is the distance measured through the centre of a cube, from one corner to its opposite corner, the cube being 3 feet, or one yard, on a side ?

Ans. 5-196 feet, nearly. We know, from the principles of geometry, that all similar surfaces, or areas, are to each other as the squares of their like dimensions.

7. Suppose we have two circular pieces of land, the one 100 feet in diameter, the other 20 feet in diameter. How much more land is there in the larger than in the smaller ?

By the above principle of geometry it follows, that the quantity of land in the two circles must be as the squares of the diameters, that is, 1002 to 20%, or as 25 to 1. Hence, there is 25 times as much in the one piece as there is in the other.

8. Suppose, by observation, it is found that 4 gallons of water flow through a circular orifice of 1 inch in diameter in 1 minute. How many gallons would, under similar circumstances, be discharged through an orifice of 3 inches in diameter, in the same length of time?'

Ans. 36 gallons. 9. What length of thread is required to wind spirally around a cylinder, 2 feet in circumference and 3 feet in length, so as to go but once around ?

It is evident that if the cylinder be developed, or placed upon a plane, and caused to roll once over, that the convex surface of the cylinder will give a rectangle, whose width is 2 feet, and length 3 feet; at the same time the thread will form its diagonal. Hence, the length of the thread is v4+9=V13=3.60555 feet, nearly.

EXTRACTION OF THE CUBE ROOT.

134. We will first involve a number to the third power, that is, we will find the cube of that number.

Let the number be 45.

453=45 x 45 x 45=91125. But we will separate this number into parts; that is, into tens and units, ar.d show by the aid of the exponent and the symbols, how the cube of the number when thus separated is obtained.

OPERATION.

40 +5
40 +5
40? +40.5

+40.5+52
402+2 x 40.5+5°=the square of 40+5.
40 +5
40+ 2 x 402.5+ 40.52

+ 409.5+2x 40.52 +53
403+3 X 409.5+3 x 40.52 +53=cube of 40+5.
By a similar process we shall obtain

(6+8)=63+3 x 69.8+3 x 6.82 +83. That is, the cube of the sum of two numbers is, the cube of the first number, plus three times the product of the square of the first number into the second, plus three times the product of the first into the square of the second, plus the cube of the second.

If we wish the cube of the sum of three numbers, as 6+8+9, we may unite the first and second by means of a parenthesis : thus, for 6+8+9, we may make use of

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(6+8)+9, and regarding (6+8) as one number, we find,
according to the foregoing statement, that the cube of
(6+8)+9 is equal to the cube of (6+8) plus three times
the product of the square of (6+8) into 9, plus three times
the product of (6+8) into the square of 9, plus the cube
of 9. But the cube of 6+8, has already been shown to
be equal to the cube of 6, plus three times the product of
the square of 6 into 8, plus three times the product of 6
into the square of 8, plus the cube of 8. Hence the cube
of 6+8+9 is equal to the cube of 6, plus three times the
square of 6 into 8, plus three times 6 into the square of 8,
plus the cube of 8; plus three times the square of the sum
of 6 and 8 into 9, plus three times the sum of 6 and 8
into the square of 9, plus the cube of 9. And in general,
we have the cube of the sum of any number of numbers equal
to the cube of the first number, plus three times the square of
the first number into the second, plus three times the first
into the square of the second, plus the cube of the second;
plus three times the square of the sum of the first two into
the third, plus three times the sum of the first two into the
square of the third, plus the cube of the third ; plus three
times the square of the sum of the first three into the fourth,
plus three times the sum of the first three into the square of
the fourth, plus the cube of the fourth, and so on.
Thus:

(2+3)=2+3 x 22.3+3x2.39 +3?.
(5+7)=53+3 x 54.7 +3X5.72 +73...

(5+6+7)=53+3 x 59.6+3 x 5.69 +63
+3x(5+6).7+3x (5+6).72 +78.

(365)=(300+60+5)=3003+3 x 300°.60
+3x300.602 +603 +3% (300+60)4.5
+3x (300+60).52 +53.

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