what is the following quotient?\n$\frac{9 + sqrt{2}}{4-sqrt{7}}$\n$\frac{9sqrt{7}+sqrt{14}}{-3}$\n$\frac{36…

what is the following quotient?\n$\frac{9 + sqrt{2}}{4-sqrt{7}}$\n$\frac{9sqrt{7}+sqrt{14}}{-3}$\n$\frac{36 - 9sqrt{7}+4sqrt{2}-sqrt{14}}{9}$\n$\frac{36 + 9sqrt{7}+4sqrt{2}+sqrt{14}}{9}$\n$\frac{79}{9}$

what is the following quotient?\n$\frac{9 + sqrt{2}}{4-sqrt{7}}$\n$\frac{9sqrt{7}+sqrt{14}}{-3}$\n$\frac{36 - 9sqrt{7}+4sqrt{2}-sqrt{14}}{9}$\n$\frac{36 + 9sqrt{7}+4sqrt{2}+sqrt{14}}{9}$\n$\frac{79}{9}$

Answer

Explanation:

Step1: Rationalize the denominator

Multiply the numerator and denominator by the conjugate of the denominator $4 + \sqrt{7}$. $\frac{9+\sqrt{2}}{4 - \sqrt{7}}\times\frac{4+\sqrt{7}}{4+\sqrt{7}}$

Step2: Expand the numerator

$(9+\sqrt{2})(4+\sqrt{7})=9\times4+9\sqrt{7}+4\sqrt{2}+\sqrt{2}\times\sqrt{7}=36 + 9\sqrt{7}+4\sqrt{2}+\sqrt{14}$

Step3: Expand the denominator

$(4 - \sqrt{7})(4+\sqrt{7})=4^{2}-(\sqrt{7})^{2}=16 - 7=9$

Answer:

$\frac{36 + 9\sqrt{7}+4\sqrt{2}+\sqrt{14}}{9}$