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8. Find the value of (3)'.

We have (3) = fx= Hence to raise a fraction to any power, raise both numerator and denominator to the required power.

9. Find the value of (a), (6), (1), (2), (3). 10. Find the value of (.4), (4), (5), (12), (3) 11. Find the value of 52 x 53.

We have 52 x 53

5 x 5 times 5 x 5 x 5

55.

Hence the product of any two powers of a number equals a power of the number denoted by the sum of the exponents. What power is

12. 82 X 83? 16. (3) X (3)'? 20. 610 x 612? 13. 73 x 72? 17. 42 x 43 x 45 ? 21. 122 X 123 ? 14. 9' X 95? 18. 22 X 28 X 24? 22. 18 X 186? 15. 35 x 36? 19. 53 x 53 x 53 ? 23. 203 X 208

24. What power is (4)?

We have (42)3 = 42 x 42 x 42 = 46. Hence a power of a number raised to any power equals å power of the number denoted by the product of the exponents. What power is25. (24) ? 29. (5%)"?

33. (12)"?
26. (3)°?
30. (73)*?

34. (154)*?
27. (42)* ? 31. (63) ?

35. (20%)*?
28. (5')??
32. (89)*?

36. (306)?

NOTE.—By means of this principle we may abbreviate the process of involution. Thus,

We may raise 5 to the 6th power by squaring its cube, or to the 9th power by cubing its cube, etc.

56 (53)2 (125)2 = 15625.

5o = (53)3 = (125)3 – 1953125.

WRITTEN EXERCISES.

Find the value of 1. 15.

10. 993 2. 45%.

11. 673. 3. 83.

12. 2.5%. 4. 992 13. 652. 5. 305. 14. 343. 6. 123.

15. (3.5). 7. 25%.

16. (24) 8. 85

17. (63)? 9. 999.

18. (33)!.

19. .992.
20. .093.
21. .15.
22. .001.
23. .9993.
24. .001.
25. .999%.
26. 75.
27. 125.

28. .0001%. 29. 2182. 30. 3622. 31. 6.05. 32. 42.12. 33. 2.37%. 34. 1001 35. 10.01. 36. 1.001.

G

52

F

SQUARING NUMBERS GEOMETRICALLY. 37. Square 25. 25 equals 20 + 5.

20 X 5 Let the line A H be 20 units long, E and H B 5 units long.

Construct a square on A H and one on A B, as indicated in the figure. Then

202 A BCD will represent the square of 25, and will consist of

20 X 5

20% + 2 x (20 x 5) + 52

400 + 200 + 25 625.

20

H 5

B

That is, the square of any number composed of two figures is equal to the square of the tens, plus twice the product of the tens into the units, plus the square of the units. Draw a figure and square the following numbers: 38. 32. 41. 53. 44. 84.

47. 245. 39. 45. 42. 63.

45. 92. 48. 357. 40. 48. 43. 72.

46. 123. 49, 582

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PROCESS.

203 3 x 202 X 5 3 x 20 x 52

53 253

Let A represent a cube 20 units on a side. 8000 Its contents are 203. 6000

Let D represent an addition to one side of 1500

the cube. Its contents are 20 x 20 x 5, or 125

202 X 5. It will be seen that there are three 15625

of these blocks, whose contents are 3 ~ 202 5.

Let E represent a corner piece which is 20 * 5 * 5, or 20 5. It is seen that there are three of these, whose contents are 3 x 20 x 52.

Finally, a small cube, O, which is 5 units on a side and whose contents are 53, will complete the new cube. We shall then have

203 + 3(202 x 5) + 3(20 x 52) + 55,

or 15625 for the contents of a cube whose side is 25.

Hence, the cube of any number consisting of two figures is equal to the cube of the tens, plus three times the tens squared into the units, plus three times the tens into the units squared, plus the units cubed. Cube the following numbers geometrically: 51. 12. 5.4. 24. 57. 54.

60. 84. 52. 15. 55. 35. 58. 67.

61. 92. 53. 18. 56. 43. 59. 72.

62. 95.

EVOLUTION.

214. Evolution is the process of finding a root of a number.

215. A Root of a number is one of the equal factors of that number.

A Root is indicated by the symbol V, called the Radical Sign.

Thus, 25 =

5 X 5, hence V25 = 5, the square root of 25. 64 4 X 4 X 4, hence y 64 = 4, the cube root of 64. 81 = 3 X 3 X 3 X 3, hence 81 = 3, the fourth root of 81,

216. The Index of the root is the number written in the angle of the radical sign to indicate what root of the number is required.

3

217. A Perfect Power is a number of which the required root can be found exactly.

Thus, 25 is a perfect square, since 52 = 25.
And 27 is a perfect cube, since 38 = 27.

[blocks in formation]

3

3

3

3

1. Of V100. 2. Of V121. 3. Of V 169. 4. Of V 144. 5. Of 1000. 6. Of V.09. 7. Of V.16. 8. Of V.25. 9. Of 8000. 10. Of V.36. 11. Of 27000. 12. Of V.49.

13. Of V400.
14. Of 64000.
15. Of 1.008.
16. Of V 4900.
17. Of .027
18. Of 512000.
19. Of v 900.
20. Of \/729000.
21. Of.343.
22. Of V.81.
23. Of v.064.
24. Of V.64.

25. Of V.125.
26. Of v 343000.
27. Of V 8100.
28. Of v.512.
29. Of V 6400.
30. Of y.729.
31. Of 1.216.
32. Of V 1600.
33. Of v 125000.
34. Of v 2500.
35. Of | 216000.
36. Of V 3600.

3

37. Find the value of it.

PROCESS.

°125 512 512

8 125

Ans.

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