Simplify the following:
(a-1 + b-1) ÷ (a-2 - b-2)
39 Likes
Given,
⇒(a−1+b−1)÷(a−2−b−2)⇒(1a+1b)÷(1a2−1b2)⇒(b+aab)÷(b2−a2a2b2)⇒(b+aab)×(a2b2b2−a2)⇒ab(b+a)b2−a2⇒ab(b+a)(b−a)(b+a)⇒abb−a.\Rightarrow (a^{-1} + b^{-1}) ÷ (a^{-2} - b^{-2}) \\[1em] \Rightarrow \Big(\dfrac{1}{a} + \dfrac{1}{b}\Big) ÷ \Big(\dfrac{1}{a^2} - \dfrac{1}{b^2}\Big) \\[1em] \Rightarrow \Big(\dfrac{b + a}{ab}\Big) ÷ \Big(\dfrac{b^2 - a^2}{a^2b^2}\Big) \\[1em] \Rightarrow \Big(\dfrac{b + a}{ab}\Big) \times \Big(\dfrac{a^2b^2}{b^2 - a^2}\Big) \\[1em] \Rightarrow \dfrac{ab(b + a)}{b^2 - a^2} \\[1em] \Rightarrow \dfrac{ab(b + a)}{(b - a)(b + a)} \\[1em] \Rightarrow \dfrac{ab}{b - a}.⇒(a−1+b−1)÷(a−2−b−2)⇒(a1+b1)÷(a21−b21)⇒(abb+a)÷(a2b2b2−a2)⇒(abb+a)×(b2−a2a2b2)⇒b2−a2ab(b+a)⇒(b−a)(b+a)ab(b+a)⇒b−aab.
Hence, (a-1 + b-1) ÷ (a-2 - b-2) = abb−a.\dfrac{ab}{b - a}.b−aab.
Answered By
25 Likes
(xaxb)a2+ab+b2.(xbxc)b2+bc+c2.(xcxa)c2+ac+a2\Big(\dfrac{x^a}{x^b}\Big)^{a^2 + ab + b^2}.\Big(\dfrac{x^b}{x^c}\Big)^{b^2 + bc + c^2}.\Big(\dfrac{x^c}{x^a}\Big)^{c^2 + ac + a^2}(xbxa)a2+ab+b2.(xcxb)b2+bc+c2.(xaxc)c2+ac+a2
(xax−b)a2−ab+b2.(xbx−c)b2−bc+c2.(xcx−a)c2−ca+a2\Big(\dfrac{x^a}{x^{-b}}\Big)^{a^2 - ab + b^2}.\Big(\dfrac{x^b}{x^{-c}}\Big)^{b^2 - bc + c^2}.\Big(\dfrac{x^c}{x^{-a}}\Big)^{c^2 - ca + a^2}(x−bxa)a2−ab+b2.(x−cxb)b2−bc+c2.(x−axc)c2−ca+a2
11+am−n+11+an−m\dfrac{1}{1 + a^{m - n}} + \dfrac{1}{1 + a^{n - m}}1+am−n1+1+an−m1
Prove the following:
(a + b)-1(a-1 + b-1) = 1ab\dfrac{1}{ab}ab1
