rationalize the denominator of

1/(2+5√3)

[1/(2+5sqrt3)](2-5sqrt3)/(2-5sqrt3)

=(2-5sqrt3)/(4-75)

= (5 sqrt3-2)/71

To rationalize the denominator of the fraction 1/(2+5√3), we need to get rid of the square root in the denominator. Here's how you can do it:

Step 1: Multiply both the numerator and denominator by the conjugate of the denominator. The conjugate of 2+5√3 is 2-5√3.

1/(2+5√3) * (2-5√3)/(2-5√3)

Step 2: Simplify the denominator by multiplying using the FOIL method (First, Outer, Inner, Last).

(2 * 2) + (2 * -5√3) + (-5√3 * 2) + (-5√3 * -5√3)
4 - 10√3 - 10√3 + 75
79 - 20√3

Step 3: Rewrite the fraction using the simplified denominator.

1/(2+5√3) * (2-5√3)/(79-20√3)

So, the rationalized denominator of the fraction 1/(2+5√3) is (79 - 20√3).