How do you rationalize $\frac{6 - \sqrt{7}}{5 - \sqrt{7}}$ $(6-sqrt7)/(5-sqrt7)$ ?

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Answer 1 · 2015-06-01T21:04:05

Multiply both numerator (top) and denominator (bottom) by the conjugate $5 + \sqrt{7}$ $5 + sqrt(7)$ ...

$\frac{6 - \sqrt{7}}{5 - \sqrt{7}}$ $(6-sqrt(7))/(5-sqrt(7))$

$= \frac{6 - \sqrt{7}}{5 - \sqrt{7}} \cdot \frac{5 + \sqrt{7}}{5 + \sqrt{7}}$ $= (6-sqrt(7))/(5-sqrt(7))*(5+sqrt(7))/(5+sqrt(7))$

$= \frac{(6 - \sqrt{7}) (5 + \sqrt{7})}{(5 - \sqrt{7}) (5 + \sqrt{7})}$ $= ((6-sqrt(7))(5+sqrt(7))) / ((5-sqrt(7))(5+sqrt(7)))$

$= \frac{30 + \sqrt{7} - {\sqrt{7}}^{2}}{5^{2} - {\sqrt{7}}^{2}}$ $= (30+sqrt(7)-sqrt(7)^2)/(5^2-sqrt(7)^2)$

$= \frac{30 + \sqrt{7} - 7}{25 - 7}$ $= (30+sqrt(7)-7)/(25-7)$

$= \frac{23 + \sqrt{7}}{18}$ $=(23+sqrt(7))/18$

How do you rationalize (6-sqrt7)/(5-sqrt7)6−√75−√7(6-sqrt7)/(5-sqrt7)?