If \(x=4-\sqrt7\), what is \(\frac1x\)?

Correct answer: \(\frac{4+\sqrt7}{9}\)

The governing method is rationalisation using the conjugate of the denominator. Starting from 1/x = 1/(4 − √7), multiply numerator and denominator by 4 + √7. The denominator becomes (4 − √7)(4 + √7) = 4² − (√7)² = 16 − 7 = 9, while the numerator is 4 + √7. Thus 1/x = (4 + √7)/9, so option A is correct. Option B gives the numerator but omits division by 9. Option C incorrectly keeps the original sign instead of using the conjugate. Option D multiplies by 9 rather than dividing by the rationalised denominator. Multiplying the proposed answer by x gives [(4 + √7)(4 − √7)]/9 = 9/9 = 1.