Show that : 1 1 + a P − Q + 1 1 + a Q − P

Show that : `(1)/(1 + "a"^("p"- "q")) + (1)/(1 + "a"^("q"- "p")`

Sum

L.H.S.

= `(1)/(1 + "a"^("p"- "q")) + (1)/(1 + "a"^("q"- "p")`

= `(1 + "a"^("q"- "p") + 1 + "a"^("p" - "q"))/((1 + "a"^("p"-"q"))(1 + "a"^("q"-"p"))`

= `(2 + "a"^-("p" -"q") + "a"^("p" - "q"))/((1 + "a"^("p"-"q"))(1 + "a"^-("q"-"p"))`

= `(2 + "a"^-("p"-"q") + "a"^("p"-"q"))/(1 + "a"^-("p"-"q") + "a"^("p"-"q") + "a"^("p"-"q") . "a"^-("p"-"q")`

= `(2 + "a"^-("p"-"q") + "a"^("p"-"q"))/(1 + "a"^-("p"-"q") + "a"^("p"-"q") + "a"^("p"-"q"-"p"+"q")`

= `(2 + "a"^-("p"-"q") + "a"^("p"-"q"))/(1 + "a"^-("p"-"q") + "a"^("p"-"q") + "a"^0`

= `(2 + "a"^-("p"-"q") + "a"^("p"-"q"))/(1 + "a"^-("p"-"q") + "a"^("p"-"q") + 1`

= `(2 + "a"^-("p"-"q") + "a"^("p"-"q"))/(2 + "a"^-("p"-"q") + "a"^("p"-"q"))`
= 1
= R.H.S.
Hence proved.

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Solving Exponential Equations

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Chapter 6: Indices - Exercise 9.1

Chapter 6 Indices
Exercise 9.1 | Q 15