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) From these three examples may be collected the following rule for the multiplication of algebraic quantities: Multiply each term of the multiplicand by each term of the multiplier; when the two terms have both + or both-before them, put + before their product; when one has + and the other-, put-before their product. In using the first terms, which have no sign, apply the rule as if they had the sign +. 68. For example, (_a_ + _b_)(_a_ + _b_) gives _aa_ + _ab_ + _ab_ + _bb_. But _ab_ + _ab_ is 2_ab_; hence the _square_ of _a_ + _b_ is _aa_ + 2_ab_ + _bb_. Again (_a_- _b_)(_a_-_b_) gives _aa_-_ab_-_ab_ + _bb_. But two subtractions of _ab_ are equivalent to subtracting 2_ab_; hence the _square_ of _a_- _b_ is _aa_-2_ab_ + _bb_. Again, (_a_ + _b_)(_a_-_b_) gives _aa_ + _ab_-_ab_ -_bb_. But the addition and subtraction of _ab_ makes no change; hence the product of _a_ + _b_ and _a_- _b_ is _aa_-_bb_.