33. A cask contains 42 U. S. gallons of wine, worth $44 per gallon. How much less will it cost in U. S. money, at the rate of £1 2s. per the Imperial gallon?

34. A garden 400 ft. long and 300 ft. wide has a walk 20 ft. wide laid off from it on each side. What is the ratio between the area of the walk and the area of what remains?

35. A commission merchant in Charleston received into his store on May 1, 1875, 1000 bbl. of flour, paying as charges on the same day, freight $175.48, cartage $56.25, and cooperage $8.37. He sold out the shipment as follows: June 3, 200 bbl. @ $6.25; June 30, 350 bbl. @ $6.50: July 29, 400 bbl. @ $6.124; Aug. 6, 50 bbl. @ $6. Required, the net proceeds, and the date when they shall be accredited to the owner, allowing commission at 31%, and storage at 2 cents per week per bbl.

36. Three men engage in manufacturing. L puts in $3840 for 6 mo. ; M, a sum not specified for 12 mo.; and N, $2560 for a time not specified. L received $4800 for his capital and profits; M, $9600 for his; and N, $4160 for his. Required, M's capital and N's time.

37. My expenditures in building a house, in the year 1874, were as follows: Jan. 16, $536.78; Feb. 20, $425.36 ; March 4, $259.25; April 24, $786.36. At the last date I sold the house for exactly what it cost, interest at 6 per cent. on the money expended added, and took the purchaser's note for the amount. What was the face of

the note?

38. A man bought a farm for $6000, and agreed to pay principal and interest in 3 equal annual installments. What was the annual payment, interest being 6%?

793. 1. What is the product of 3 used twice as a factor?

2. What is the product of 3 used 3 times as factor.

3. What is the product of 4 used 3 times as a factor? 4. What is the result of using 5 twice as a factor? 5. What is the product of used twice as a factor? 6. What is the result of using twice as a factor? , three times as a factor?

7. What number will be produced by using .3 twice as a factor? .7, twice? .4, three times? .05, twice?

8. A room is 9 ft. on each side; how many square fcet in the floor?

9. A cubical block of stone is 4 ft. on each edge; how many cubic feet does it contain ?

DEFINITIONS.

794. A Power of a number is the product of factors, each of which is equal to that number. Thus, 27 is the third power of 3, since 27 = 3 × 3 × 3.

795. Involution is the process of finding any power

of a number.

796. The Base or Root of a power is one of the equal factors of the power. Thus, 27 is the third power of 3, and 3 is the base, or root, of that power.

797. The Exponent of a power is a number placed at the right of the base and a little above it, to show how many times it is used as a factor to produce the power. It also denotes the degree of the power.

31 or 3

32 = 3 x 3

Thus,

33 = 3 x 3 x 3

= 27, the third power of 3.

34 = 3 x 3 x 3 x3 = 81, the fourth power of 3.

3 = 3 x 3 = 9

798. The Square of a number is its second power, so called because the number of superficial units in a square is equal to the second power of the number of linear units in one of its sides.

83 = 3 x 3 x 3 = 27

799. The Cube of a number is its third power, so called because the number of units of volume in a cube is equal to the third power of the number of linear units in one of its edges.

800. A Perfect Power is a number which can be. resolved into equal factors. Thus, 25 is a perfect power of the second degree, and 27 is a perfect power of the third degree.

801. PRINCIPLE.-The sum of the exponents of two powers of the same number is equal to the exponent of the product of those powers. Thus, 22 × 23-25; for 22-2 × 2, and 23=2x2x2; hence 22 x 23=2 × 2 × 2 × 2 × 2=25.

WRITTEN EXERCISES.

802. To find any power of a number.

1. Find the third power of 35.

OPERATION.

35 = 351; 35 × 35 = 352 = 1225 1225 × 35 353 — 42875

ber (797), 35 × 35 × 35 = 353 = 42875.

ANALYSIS. Since using

any number three times as a factor produces the third power of that num

Of 42.

Of 56.

Of 75.

Of 42. Of 54.

2. Find the square of 37. 3. Find the cube of 15. Of 18. 4. What is the value of 632? of 483? of 324 ? of 125?

RULE. Find the product of as many factors, each equal to the given number, as there are units in the exponent of the required power.

5. What is the third power of 4?

OPERATION.-( 1 ) 3 = { × 3 × }

=

4×4 × 4

43 64

=
5x5x5 53 125'

RULE.-A fraction may be raised to any power by in volving each of its terms separately to the required power.

6. What is the square of? The cube of?

7. Raise to the 4th power. 2 to the 5th power.

Find the required power of the following:

Find the value of each of the following expressions:

FORMATION OF SQUARES AND CUBES BY THE ANALYT ICAL METHOD.

803. To find the square of a number in terms of its tens and units.

1. Find the square of 27 in terms of its tens and units.

PRINCIPLE.-The square of a number consisting of tens and units is equal to the sum of the squares of the tens and units increased by twice their product.