To simplify the expression, we need to rationalize the denominator:
7_4√3 + 1 ÷ 7_4√3
= (7_4√3 + 1) ÷ 7_4√3 (since division is the same as multiplication by the reciprocal)
= (28 + 4√3 + 1) ÷ 28 (multiplying the numerator and denominator of the first fraction by the conjugate of the denominator)
= (29 + 4√3) ÷ 28
So the simplified expression is (29 + 4√3) ÷ 28.