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प्रश्न
Prove the following:
`("a"^"m"/"a"^"n")^("m"+"n"+1) ·("a"^"n"/"a"^1)^("n" + 1-"m").("a"^1/"a"^"m")^(1+"m"-"n")`
उत्तर
L.H.S.
= `("a"^"m"/"a"^"n")^("m"+"n"+1) ·("a"^"n"/"a"^1)^("n" + 1-"m").("a"^1/"a"^"m")^(1+"m"-"n")`
= `"a"^("m"("m" + "n" - 1))/("a"^("n"("m"+"n" - 1)))·"a"^("n"("n" + 1 - "m"))/("a"^(1("n"+ 1 - "m")))·"a"^(1(1 + "m" - "n")) /"a"^("m"(1 + "m" - "n")) ` ......(Using(am)n = amn)
= `"a"^("m"^z + "mn" - "m")/"a"^("n"^z+"mn" - "n")·"a"^("n"^z - "mn" + "n")/"a"^("n"+1-"m")·"a"^(1 + "m" - "n")/"a"^("m"^z - "mn" + "m")`
= `"a"^("m"^z + "mn" - "m" - ("n"^z+"mn"-"n")) ·"a"^("n"^z - "mn" - ("n" + 1 - "m"))·"a"^(1+"m"-"n"-("m"^z-"mn"+"m"))` ....(Using am ÷ an = am-n)
= `"a"^("m"^z+"mn"-"m"-"n"^z-"mn"+"n")·"a"^("n"^z-"mn"+"n"-"n"-1+"m")·"a"^(1+"m"-"n"-"m"^z-"mn"+"m")`
= `"a"^("m"^z + "mn"-"m"-"n"^z-"mn"+"n"+"n"^z-"mn"+"n"-"n"-1+"m"+1+"m"-"n"-"m"^z+"mn"-"m")` ....(Using am x an = am+n)
= a°
= 1 .....(Using a° = 1)
= R.H.S.
Hence proved.
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