By analogy, 7 is written 71 and called the first power of 7. III.-Particular cases of Propositions which are proved by Algebraic Reasoning. The student should go through the whole of these, adding others similar to them if necessary, until he performs all such operations by habit, without rules. 106 28 = 2 X = 10 X 105 102 X 104 103 X 103 = 1000000 28 X 27 =215 84 83 = 1 = 87 - 84 The preceding operations occur perpetually in the higher applications of arithmetic; the student should repeat them on low numbers, till he is perfectly familiar with all of them. The following are the rules under which they may all be reduced; but they should be dispensed with if possible, by mere habit of performing the operations. Rule 1. All roots may be treated as powers, that is, fall under the same rules as powers, when the fraction which has the order of the root in its denominator is used as the exponent. Call them fractional powers, so that the word power shall mean both power and root. IV. Various Combinations of the preceding Propositions applied to the Use of the Square-Root. Rule 1. To square the sum of two quantities, square each of them, and to the sum of the squares add twice the product of the quantities. To square the difference of two quantities, subtract twice the product, instead of adding. |