Elements of Algebra

110 ELEMENTS SECT. II. CHAP. X. *Of the higher Powers of Compound Quantities.* 340. After squares and cubes, we must consider higher powers, or powers of a greater number of degrees; which are generally represented by exponents in the manner before explained: we have only to remember, when the root is compound, to enclose it in a parenthesis: thus, $(a+b)^5$ means that $a+b$ is raised to the fifth power, and $(a-b)^6$ represents the sixth power of $a-b$, and so on. We shall in this chapter explain the nature of these powers. 341. Let $a+b$ be the root, or the first power, and the higher powers will be found, by multiplication, in the following manner: $(a+b)^1=a+b$ $a+b$ $a+b$ ----------- $a^2+ab$ $\quad ab+b^2$ ----------- $(a+b)^2=a^2+2ab+b^2$ $a+b$ ----------- $a^3+2a^2b+ab^2$ $\quad a^2b+2ab^2+b^3$ ----------- $(a+b)^3=a^3+3a^2b+3ab^2+b^3$ $a+b$ ----------- $a^4+3a^3b+3a^2b^2+ab^3$ $\quad a^3b+3a^2b^2+3ab^3+b^4$ ----------- $(a+b)^4=a^4+4a^3b+6a^2b^2+4ab^3+b^4$ $a+b$ ----------- $a^5+4a^4b+6a^3b^2+4a^2b^3+ab^4$ $\quad a^4b+4a^3b^2+6a^2b^3+4ab^4+b^5$ ----------- $a^5+5a^4b+10a^3b^2+10a^2b^3+5ab^4+b^5$