Ratienalise \( \frac{5}{9-4 \sqrt{2}} \)

Sure, let's rationalize the denominator of the expression \( \frac{5}{9-4\sqrt{2}} \). To do this, we need to multiply both the numerator and the denominator by the conjugate of the denominator. The conjugate of \( 9-4\sqrt{2} \) is \( 9+4\sqrt{2} \). So, we have: \[ \frac{5}{9-4\sqrt{2}} \times \frac{9+4\sqrt{2}}{9+4\sqrt{2}} \] Now, let's multiply the numerators and the denominators: Numerator: \[ 5 \times (9+4\sqrt{2}) = 45 + 20\sqrt{2} \] Denominator: \[ (9-4\sqrt{2}) \times (9+4\sqrt{2}) = 9^2 - (4\sqrt{2})^2 = 81 - 32 = 49 \] So, the rationalized expression is: \[ \frac{45 + 20\sqrt{2}}{49} \] This is the final answer.