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1. From a vessel on the ocean, a light in a certain lighthouse could be seen dimly at a distance of 20 miles. How much brighter would the light appear, when the vessel was 5 miles from the lighthouse?

Ans. 16 times as bright.

2. Having two lamps, one of 4 candle power and one of 16 candle power, if the former is 10 ft. distant, how far away must I place the latter to give me the same amount of light? Ans. 20 ft. 3. A bell heard by A on a certain street is heard by B 6 times as far distant. How loud will it sound to B as compared to A? Ans. as loud. 4. If the earth were removed to its present distance from the sun, how much more intense would be the heat received by it? Ans. 16 times.

5. A body 4000 miles from the center of the earth (or at the earth's surface) weighs 900 pounds. What would it weigh 12000 miles from the center of the earth? Ans. 100 pounds.

6. Two magnetic poles inch apart have an attraction for each other whose force would lift a pound weight. What weight would they lift if they were an inch apart?




7. A pistol shot is heard by A. To B it sounds loud. How much further is B from the pistol than A? 8. The lights on a certain tower appear very bright to A. They appear as bright to B. Other things being equal, how much further is B from the light than A?

9. A wax taper is held a certain distance from a flame. It requires an amount of heat 16 times as great as that now acting upon it to ignite it. How much nearer must I move the taper if I wish to light it?


426. A Root is a factor repeated to produce a power.

Thus, in the expression 5 × 5 × 5 = 125, 5 is the root from which the power, 125, is produced.

427. Evolution is the process of extracting the root of a number considered as a power, and is the reverse of Involution.

428. The Radical Sign is the character, √, which, placed before a number, denotes that its root is to be extracted.

429. The Index of the root is the figure placed above the radical sign, to denote what root is to be taken. When no index is written, the index 2 is always understood.

430. A Surd is the indicated root of an imperfect power.

431. Roots are named from the corresponding powers, as will be seen in the following illustrations: The square root of 9 is 3, written √9-3.


The cube root of 27 is 3, written 27-3.
The fourth root of 81 is 3, written √81=3.

Any number whatever may be considered a power whose root is to be extracted; but only the perfect powers can have exact roots.



432. To find any root of a number by factoring.

1. Find the cube root of 1728.

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- A number that is a perfect cube is composed of three equal factors, and one of them is the cube root of that number.

The prime factors of 1728 are 3, 3, 3, 2, 2, 2, 2, 2, 2 hence 1728 (3 × 2 × 2) × (3 × 2 × 2) × (3 × 2 × 2); therefore the cube root of 1728 is (3 × 2 × 2),

or 12.



RULE. - Resolve the given number into its prime factors; then, to produce the square root, take one of every two equal factors; to produce the cube root take one of every three equal factors; and so on.

2. Find the square root of 64, 256, 576, 6561.
3. Find the cube root of 729, 2744, 9261, 3375.
4. Find the square root of 225, 256, 289.
5. Find the cube root of 4913, 8000, 24389.
6. Find the fourth root of 81, 256, 625, 1296.
7. Find the fifth root of 243, 1024, 3125.
8. Find the sixth root of 729, 4096, 15625.
9. Find the seventh root of 2187, 16384, 78125.
10. Find the square root of 8836, 2704, 1444.

11. Find the cube root of 2197, 42875, 1728.

12. Find the square root of 529, 5184, 192721, 1521, 6889.

13. Find the cube root of 6859, 2744, 1331.

14. Find the square root of 1089, 14161, 156025.


433. The Square Root of a number is one of the two equal factors that produce the number.

Thus, the square root of 49 is 7, for 7 x 7 = 49.

In extracting the square root, the first thing to be determined is the relative number of places in a given number and its square root. The law governing this relation is exhibited in the following:

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From these examples we perceive

1. That a root consisting of 1 place may have 1 or

2 places in the square.

2. That in all cases the addition of 1 place to the root adds 2 places to the square. Hence,

If we point off a number into two-figure periods, commencing at the right hand, the number of full periods and the left hand full or partial period will indicate the number of places in the square root; the highest period corresponding to the highest figure of the root.

To ascertain the relations of the several figures of the root to the periods of the number, observe that if any number, as 2345, is decomposed, the squares of the left hand parts will be related in local value as follows:

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The square of the first figure of the root is contained wholly in the first period of the power; the square of the first two figures of the root in the first two periods of the power; and so on.


434. 1. What is the length of one side of a square plot containing an area of 5417 sq. ft.?


54,17 | 73.6


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140 517

146.0 88.00
146.6 87.96

SOLUTION. Since the given figure is a square, its side will be the square root of its area, which we will proceed to compute. Pointing off the given number, the two periods show that there will be two integral figures, tens and units, in the root. The tens of the root must be extracted from the first or left hand period, 54 hundreds. The greatest square in 54 hundreds is 49 hundreds, the square of 7 tens; we therefore write 7 tens in the root, at the right of the given number. Since the entire root is to be the side of a square, let us form a square (Fig. I), the side of which is 70 feet long. The area of this square is 70 × 70 = 4900 sq. ft., which we subtract from the given number. This is done in the operation by subtracting the square number, 49, from the first period, 54, and to the remainder bringing down the second period, making the entire remainder 517.


Fig. 1.


If we now enlarge our square (Fig. I) by the addition of 517 square feet, in such a manner as to preserve the square form, its size will be that of the required square. To preserve the square form, the addition must be so made as to extend

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the square equally in two directions; it will therefore be composed of two oblong figures at the sides and a little square at the corner (Fig. II). Now, the width of this addition will be the additional length to the side of the square, and consequently the next figure in the root. To find width we divide square contents, or area, by length. But the length of one side of the little square cannot be found till the width of the addition is determined, because it is equal to this width. We will therefore add the lengths of the two oblong figures, and the sum will be sufficiently near the whole length to be used as a trial divisor.

Fig. II.

Each of the oblong figures is equal in length to the side of the square first formed; and their united length is 70+ 70 140 ft.


PRAC. AR.- - 23

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