8) Rationalize the Denominator: (7)/(9-sqrt (2))

To rationalize the denominator of the expression $\frac{7}{9-\sqrt{2}}$, we need to eliminate the square root in the denominator. We can do this by multiplying both the numerator and denominator by the conjugate of the denominator, which is $9+\sqrt{2}$.<br /><br />The conjugate of a binomial expression $a-b$ is $a+b$. In this case, the conjugate of $9-\sqrt{2}$ is $9+\sqrt{2}$.<br /><br />So, we multiply the numerator and denominator by $9+\sqrt{2}$:<br /><br />$\frac{7}{9-\sqrt{2}} \cdot \frac{9+\sqrt{2}}{9+\sqrt{2}}$<br /><br />Now, we can simplify the expression by multiplying the numerators and denominators:<br /><br />$\frac{7(9+\sqrt{2})}{(9-\sqrt{2})(9+\sqrt{2})}$<br /><br />Next, we can simplify the denominator by using the difference of squares formula: $(a-b)(a+b) = a^2 - b^2$. In this case, $a = 9$ and $b = \sqrt{2}$.<br /><br />$(9-\sqrt{2})(9+\sqrt{2}) = 9^2 - (\sqrt{2})^2 = 81 - 2 = 79$<br /><br />So, the expression becomes:<br /><br />$\frac{7(9+\sqrt{2})}{79}$<br /><br />Finally, we can simplify the numerator by distributing the 7:<br /><br />$\frac{63+7\sqrt{2}}{79}$<br /><br />Therefore, the rationalized form of the expression $\frac{7}{9-\sqrt{2}}$ is $\frac{63+7\sqrt{2}}{79}$.