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Question
Prove that `2(bar"a" - bar"b") xx 2(bar"a" + bar"b") = 8(bar"a" xx bar"b")`
Solution
LHS = `2(bar"a" - bar"b") xx 2(bar"a" + bar"b")`
`= 4[(bar"a" - bar"b")xx(bar"a" + bar"b")]`
`= 4[bar"a" xx (bar"a" + bar"b") - bar"b" xx (bar"a" + bar"b")]`
`= 4 [bar"a" xx bar"a" + bar"a" xx bar"b" - bar"b" xx bar"a" - bar"b" xx bar"b"]`
= `4[bar0 + bar"a" xx bar"b" + bar"a" xx bar"b" - bar0]` ....`[∵ bar"a" xx bar"a" = bar"b" xx bar"b" = bar"0" "and" - (bar"b" - bar"a") = bar"a" xx bar"b"]`
= `4[2(bar"a" xx bar"b")]`
`= 8 (bar"a" xx bar"b")`
= RHS
∴ `2(bar"a" - bar"b")xx2(bar"a" + bar"b") = 8(bar"a" xx bar"b")`
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