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49. How much wood in a pile 96 ft. 6 in. long, 4 feet wide, if of the length is 3 st. 3 in. high, and #, 3 ft. 10 in. and the remaining part 4 ft. ' Ans. 1 1 C. 42 ft. 2 in. 2". 50. At 9 s. 7d. for 3 lb. of coffee, what will # of 2 lb. come to ? Ans. 4 s. 3; d. 51. At 134 cents for ; of a pound of cinnamon, what will 4 lb. 5 oz. come to ? Ans. $2.30. 52. A has of a yard of broadcloth for which he gave at the rate of $8.4 per yard. He gives the broadcloth and 50 cents for 14 yards of cassimere. What did the cassimere cost him per yard 2 Ans. $2,663. 53. A gentleman was 6 days travelling from Boston to Washington city. He paid 63 cents per mile

for stage fare, and $1,75 a day for other expenses. .

His whole expense was $37,75; what is the distance between the two places ! o Ans. 436 miles. 54. A gentleman left his estate to be divided between his widow, two sons and a daughter as follows: the widow’s share was to equal the elder son's share plus $525; the elder son’s to equal the younger son's plus the daughter's ; the younger son's to equal ; of ; of the whole estate; and the daughter's share to equal # of the younger son's share. What was the share of each, and the amount of the whole estate 2 See Rule, page 175. Widow's share, $1575. | Elder son’s “ $1050. Ans.< Younger son’s “ $735. Daughter’s “ $315. Whole estate, $3675. 55. At 6 o'clock the minute and hour hands of a clock point in opposite directions; how long before they will first point in the same direction. Ans. 321°r minutes.

SECTION IX.

INVOLUTION.
DEFINITIONS,

.1 Root is a number from which any power of that number may be obtained by being multiplied inte itself a certain number of times. A Power is the last product of a root multiplied into itself a given number of times. [If 5 be multiplied by itself, 5 is the root and 25 the 2d power.] An Index is a small figure or figures set to the right, and a little above the root, to denote what power is required. [Thus, 21* signifies the 2d power of 21.] A Line is length without breadth or thickness. .4 Right, or Straight Line is a line which does not - change its direction, or it is A B the shortest distance between two points; as A B. An Angle is the opening of two lines when they meet in the same point, or it is the space included between them when they proceed from the same point, as C D E. When an angle is read, the letter standing at the point where the lines meet, is read in the middle, as C D E, or E D C. A Perpendicular Line is one which forms a right - angle with another line. [If the angle A C B be equal to the angle A CD, the line A C is perpendicular to the line B D, and the angles A C B -D and A C D are right an

gles.]

ID

Parallel Lines have the same perpendicular dis

tance between them in

a-'b every part, as a b and o c d.

.# Triangle is a figure having three sides. If one of the angles in a trianF gle be a right angle, the *† triangle is called a right angled triangle. F G

H is a triangle.

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.4 Parallelogram is a figure of four sides, having

the opposite sides parallel. If T K I K be parallel to M L. and K F--1 L to I M, the figure I K L M is a parallelogram. When the angles in a parallelogram are right ones, it is called a rectan

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.4 Square is a figure having 4 equal sides, and 4 right angles. If O P be equal to, or as long as each of the other sides, and the angles Q O P, O PR, PR Q, and R Q O, be right angles, the figure O P RQ is a square.

Involution is the method of finding a required power from a given root.

Remark.-The 2d power of a number is generally called the square, and the third power, the cube of that number.

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- RULE,
For finding any power of a given root.

Multiply the given root into itself as many times as the index has units wanting one, the last product will be the power required.

Note—The reason for multiplying once less than the number of units in the index, is because the root itself, or given number, is called the first power, and the first product must be the 2d, and the

2d product, the third power,” &c.

1. 2.

3. 4.

5. 6. 7. 8. 9.

10. 11.

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Ans. 4096.

The 204 is how much 3 Ans. 160000.
What is the square of 1 ? Ans. 1.
The 3° is how much Ans. 27.
What number is equal to 7° 7 Ans. 49.
9° is equal to what number ? Ans. 729.
The 5* is how much 7 Ans. 3125.

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15. What is the difference between 373 and 454 3

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16. What is the product of 13° multiplied by 13°

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17. What is the 5th power of 13 *

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18. What is the product of the 2° into the 2° 2

* The power of a number is so called because it is what the roet

Ans. 2048.

is just capable of producing, when multiplied into itself.

19. What is the 11th power of 2? Ans. 2048.

In the 16th question, we multipled the third power of 13 by the square of 13; in the 17th question, we raised 13 to the 5th power, and found the same number as when we multiplied the 3d power of 13 by the 2d power of 13. In the 18th question we multiplied the 3d power of 2 by the 8th power of 2, and obtained the same number, as when we raised 2 to the 11th power in the 19th question. By examining these four questions, it will be seen that multiplying two powers of the same number together, we shall obtain that power of the same number, or root, whose index is equal the sum of the indices of those powers multiplied into themselves. In the 16th duestion, the sum of the indices is 5, and in the 17th question, the given index is 5;—in both these questions we obtain the 5th power of 13. The sum of the indices in the 18th question, is 11, a number equal to the given index in the 19th, and in both questions we find the 11th power of 2. This principle will hold true in all cases, for multiplying one power by another, we increase the power multiplied as much faster than we should were we to multiply by the root itself, as the multiplier is greater than the root. If the 4th power of 4 be required, we may obtain it by multiplying 4 three times into itself, as in the first operation below, or by multiplying the 2d power of 4, which is 16, by 16, as in the 2d operation.

Operation 1. Operation 2.
4 1 6
4 1 6
H 6 9 6
4 1 6
6 4 2 5
4.

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