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Question
Simplify `(173 xx 173 xx 173 xx 127 xx 127 xx 127)/(173 xx 173 xx 173 xx 127 xx 127 xx 127)`
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Solution
`(173 xx 173 xx 173 xx 127 xx 127 xx 127)/(173 xx 173 xx 173 xx 127 xx 127 xx 127)`
`= (173^2 + 127^3)/(173^2 - 173 xx 127 + 127^2)`
`= ((173 + 127)(173^3 - 173 xx 127 + 127^2))/(173^2 - 173 xx 127 + 127^2)` `[∵ a^3 - b^3 = (a - b)(a^2 + ab + b^2)]`
`= (a - 1/a)(a^2 + 1 + 1/a^2) - 2(a - 1/a)`
`= (a - 1/a)(a^2 + 1 + 1/a^2 - 2)`
`= (a - 1/a)(a^2 + 1/a^2 - 1)`
`∴ a^3 - 1/a^3 - 2a + 2/a = (a - 1/a)(a^2 + 1/a^2 - 1)`
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