ationalise \( \Rightarrow 9-4 \sqrt{2} \)

To rationalize the expression \( 9 - 4\sqrt{2} \), we need to eliminate the square root from the denominator. However, in this case, there is no denominator, so we are not dealing with a fraction. Instead, we are trying to rationalize the expression by removing the square root from the term itself. To rationalize \( 9 - 4\sqrt{2} \), we can multiply the expression by its conjugate. The conjugate of \( 9 - 4\sqrt{2} \) is \( 9 + 4\sqrt{2} \). When we multiply the original expression by its conjugate, we get: \[ (9 - 4\sqrt{2})(9 + 4\sqrt{2}) \] This is a difference of squares, which can be simplified as follows: \[ a^2 - b^2 = (a - b)(a + b) \] In this case, \( a = 9 \) and \( b = 4\sqrt{2} \), so we have: \[ (9 - 4\sqrt{2})(9 + 4\sqrt{2}) = 9^2 - (4\sqrt{2})^2 \] Now we calculate the squares: \[ 9^2 = 81 \] \[ (4\sqrt{2})^2 = 4^2 \cdot (\sqrt{2})^2 = 16 \cdot 2 = 32 \] Subtract the second square from the first: \[ 81 - 32 = 49 \] So, the rationalized form of \( 9 - 4\sqrt{2} \) is simply \( 49 \). However, this is not a rationalization in the traditional sense, as we did not remove the square root from the original expression; instead, we squared the expression to eliminate the square root. If you were looking to rationalize the denominator of a fraction that contained \( 9 - 4\sqrt{2} \), you would multiply by the conjugate as shown above.