+296 √53+20√7 can be written in the form a+b√c, where a,b and c are integers and c has no factors which is a perfect square of any positive integer other than 1. Find a+b+c.
+130477 √ [53 + 20√ 7] = a + b√ c square both sides
53 + 20√ 7 = a^2 +2ab√ c + b^2c
Equating terms
2ab = 20 a^2 + b^2c = 53
ab =10 a^2 + 7b^2 =53
And c = 7
Then
a^2 + 7b^2 = 53
Let b = 2
a^2 + 7(2)^2 = 53
a^2 + 28 = 53
a^2 = 25
a = 5
So
a + b√ c = 5 + 2√ 7
So
a + b + c = 5 + 2 + 7 = 14
