Without expanding, show that the value of each of the following determinants is zero: `|(sqrt(23)+sqrt(3),sqrt(5),sqrt(5)),(sqrt(15)+sqrt(46),5,sqrt(10)),(3+sqrt(115),sqrt(15),5)|` .

Explanation

We have
` Delta=|{:(sqrt(13)+sqrt(3),,2sqrt(5),,sqrt(5)),(sqrt(15)+sqrt(26),,5,,sqrt(10)),(3+ sqrt(65),,sqrt(15),,5):}|`
Taking `sqrt(5)` common from`C_(2) " and " C_(3)` we get
` Delta =(sqrt(5))^(2) |{:(sqrt(13)+sqrt(3),,2,,1),(sqrt(15)+sqrt(26),,sqrt(5),,sqrt(2)),(3+ sqrt(65),,sqrt(3),,sqrt(5)):}|`
Applying `C_(1) to C_(1) -sqrt(3)C_(2) -sqrt(13)C_(3)` we get
`|{:(-sqrt(3),,2,,1),(0,,sqrt(5),,sqrt(2)),(0,,sqrt(3),,sqrt(5)):}|`
(Expanding along `C_(1))`