Rationalization method is to be used while solving sums like these
Rationalization factor for :
1) 1/√a ------> √a
2) a + √b -------> a - √b
3) a - √b ---------> a + √b
4) √a + √b ----------> √a - √b
5) √a - √b ------------> √a + √b

699 / (√289 + √25)
As seen in above Rationalization Methods we can use 4^th rule to solve the problem
Thus 699 / (√289 + √25) is multiplied the rationalizing factor i.e. √289 - √25
To balance the extra added rationlizing factor we multiply the given ratio by √289 - √25 / √289 - √25
Thus the result of (699 / (√289 + √25)) * (√289 -√25 / √289 - √25) = (699 * (√289 - √25) / ((√289)² - (√25)²) [ use the identity of (a+b)(a-b) = a²- b²]
i.e. 699 / √289 + √25 = (699 / √289 + √25) * (√289 -√25 / √289 - √25) = (699 * (√289 - √25) / (289 - 25)
i.e. 699 / √289 + √25 = (699 / √289 + √25) * (√289 -√25 / √289 - √25) = (699 * (12) / (289 - 25)
i.e. 699 / √289 + √25 = (699 / √289 + √25) * (√289 -√25 / √289 - √25) = (8388) / (289 - 25)
i.e. 699 / √289 + √25 = (699 / √289 + √25) * (√289 -√25 / √289 - √25) = (8388) / (264)
i.e. 699 / √289 + √25 = (699 / √289 + √25) * (√289 -√25 / √289 - √25) = 31.772727272727
i.e. 699 / √289 + √25 = (699 / √289 + √25) * (√289 -√25 / √289 - √25) = 32 (rounded off to nearest integer)