62. Suppose it is required to multiply 23707 by 4567. Since 4567 is made up of 4000, 500, 60, and 7, by (53) we must multiply 23707 by each of these, and add the products. Now (58) 23707 × 7 is 165949 (60) 23707 × 60 is 1422420 23707 × 500 is 11853500 23707 × 4000 is 94828000 --------- The sum of these is 108269869 which is the product required. It will do as well if, instead of writing the ciphers at the end of each line, we keep the other figures in their places without them. If we take away the ciphers, the second line is one place to the left of the first, the third one place to the left of the second, and so on. Write the multiplier and the multiplicand over these lines, and the process will stand thus: 23707 4567 ------ 165949 142242 118535 94828 --------- 108269869 63. There is one more case to be noticed; that is, where there is a cipher in the middle of the multiplier. The following example will shew that in this case nothing more is necessary than to keep the first figure of each line in the column under the figure of the multiplier from which that line arises. Suppose it required to multiply 365 by 101001. The multiplier is made up of 100000, 1000 and 1. Proceed as before, and 365 × 1 is 365 (57) 365 × 1000 is 365000 365 × 100000 is 36500000 -------- The sum of which is 36865365 and the whole process with the ciphers struck off is: 365 101001 ------ 365 365 365 -------- 36865365 64. The following is the rule in all cases: I. Place the multiplier under the multiplicand, so that the units of one may be under those of the other. II. Multiply the whole multiplicand by each figure of the multiplier (59), and place the unit of each line in the column under the figure of the multiplier from which it came. III. Add together the lines obtained by II. column by column. 65. When the multiplier or multiplicand, or both, have ciphers on the right hand, multiply the two together without the ciphers, and then place on the right of the product all the ciphers that are on the right both of the multiplier and multiplicand. For example, what is 3200 × 3000? First, 3200 is 32 × 100, or one hundred times as great as 32. Again, 32 × 13000 is 32 × 13, with three ciphers affixed, that is 416, with three ciphers affixed, or 416000. But the product required must be 100 times as great as this, or must have two ciphers affixed. It is therefore 41600000, having as many ciphers as are in both multiplier and multiplicand. 66. When any number is multiplied by itself any number of times, the result is called a _power_ of that number. Thus: