Elements of Algebra

100 $\qquad$ ELEMENTS $\qquad$ SECT. II. Positive Parts. $\qquad$ Negative Parts. $300^2 = 90000$ $\qquad$ $2(40 \times 300) = 24000$ $40^2 = 1600$ $\qquad$ $2(4 \times 300) = 2400$ $2(40 \times 4) = 320$ $\qquad$ ____________ $4^2 = 16$ $\qquad$ $\qquad \qquad - 26400$ ____________ $\qquad 91936$ $\qquad - 26400$ ____________ $\qquad 65536$, the square of 256 as before. CHAP. VII. *Of the Extraction of Roots applied to Compound Quantities.* 317. In order to give a certain rule for this operation, we must consider attentively the square of the root $a+b$, which is $a^2 + 2ab + b^2$, in order that we may reciprocally find the root of a given square. 318. We must consider therefore, first, that as the square, $a^2 + 2ab + b^2$, is composed of several terms, it is certain that the root also will comprise more than one term; and that if we write the terms of the square in such a manner, that the powers of one of the letters, as $a$, may go on continually diminishing, the first term will be the square of the first term of the root; and since, in the present case, the first term of the square is $a^2$, the first term of the root must be $a$. 319. Having therefore found the first term of the root, that is to say, $a$, we must consider the rest of the square, namely, $2ab + b^2$, to see if we can derive from it the second part of the root, which is $b$. Now, this remainder, $2ab + b^2$, may be represented by the product, $(2a + b)b$; wherefore the remainder having two factors, $(2a + b)$, and $b$, it is evident that we shall find the latter, $b$, which is the second part of the root, by dividing the remainder, $2ab + b^2$, by $2a + b$. 320. So that the quotient, arising from the division of the above remainder by $2a + b$, is the second term of the root required; and in this division we observe, that $2a$ is the double of the first term $a$, which is already determined: so that although the second term is yet unknown, and it is necessary, for the present, to leave its place empty, we may nevertheless attempt the division, since in it we attend only