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Show that the square to (sqrt(26-15))//(...

Show that the square to `(sqrt(26-15))//(5sqrt(2)-sqrt(38-5sqrt(3)))` is a rational number.

Text Solution

Verified by Experts

The correct Answer is:
`3`
`5sqrt(2) - sqrt(38 + 5sqrt(3)) = 5sqrt(2) - (1)/(sqrt(2)) sqrt(76 + 2 sqrt(75))`
`= 5sqrt(2) - (1)/(sqrt(2)) -(sqrt(75) + 1) = (sqrt(3)(3sqrt(3) - 5))/(sqrt(2))`
`:. (sqrt(26 - 15sqrt(3)))/(5sqrt(2) - sqrt(38 + 5 sqrt(3))) = (sqrt(52 - 30sqrt(3)))/(sqrt(3)(3sqrt(3) - 5))`
`= (3sqrt(3) - 6)/(sqrt(3)(3sqrt(3)-5)) = (1)/(sqrt(3))` so `p = 1 q = 3`
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