66. When any number is multiplied by itself any number of times, the result is called a power of that number. Thus:
| 6 is called the | first power | of 6 |
| 6 × 6 | second power | of 6 |
| 6 × 6 × 6 | third power | of 6 |
| 6 × 6 × 6 × 6 | fourth power | of 6 |
| &c. | &c. |
The second and third powers are usually called the square and cube, which are incorrect names, derived from certain connexions of the second and third power with the square and cube in geometry. As exercises in multiplication, the following powers are to be found.
| Number proposed. | Square. | Cube. |
|---|---|---|
| 972 | 944784 | 918330048 |
| 1008 | 1016064 | 1024192512 |
| 3142 | 9872164 | 31018339288 |
| 3163 | 10004569 | 31644451747 |
| 5555 | 30858025 | 171416328875 |
| 6789 | 46090521 | 312908547069 |
| The fifth | power of 36 is | 60466176 |
| fourth | 50 | 6250000 |
| fourth | 108 | 136048896 |
| fourth | 277 | 5887339441 |
67. It is required to multiply a + b by c + d, that is, to take a + b as many times as there are units in c + d. By (53) a + b must be taken c times, and d times, or the product required is (a + b)c + (a + b)d. But (52) (a + b)c is ac + bc, and (a + b)d is ad + bd; whence the product required is ac + bc + ad + bd; or,
(a + b)(c + d) = ac + bc + ad + bd.
By similar reasoning
(a - b)(c + d) is (a - b)c + (a - b)d; or,
(a - b)(c + d) = ac - bc + ad - bd.
To multiply a-b by c-d, first take a-b c times, which gives ac-bc. This is not correct; for in taking it c times instead of c-d times, we have taken it d times too many; or have made a result which is (a-b)d too great. The real result is therefore ac-bc-(a -b)d. But (a-b)d is ad- bd, and therefore
(a - b)(c - d) = ac - bc - ad - bd
= ac - bc - ad + bd (41)