√(a+b√3)=13/(4+√3)

Question

√(a+b√3)=13/(4+√3) where a and b are integers, find, without using a calculator

Answer 1

I usually do not look at the answer another tutor has provided, but I had worked on this problem last night and wanted to be sure it had been answered. What Mark M. did is correct, but I think he missed a part of the problem, namely, the radical at the very beginning of the problem.

As a consequence his last line should be 4 - sqrt(3) = sqrt[a + bsqrt(3)}

and the sides need to be squared in order to get the complete solution.

a + bsqrt(3) = 19 - 8sqrt(3) => a = 19 and b = -8.

Answer 2

13 / (4 + √3) = [13(4 - √3)] / [(4 + √3)(4 - √3)]

= 13(4 - √3) / 13

= 4 - √3 = a + b√3, where a = 4 and b = -1