How do you rationalize the denominator and simplify $\frac{1}{5 + \sqrt{7}}$ $1/(5+sqrt7)$ ?

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How do you rationalize the denominator and simplify $\frac{1}{5 + \sqrt{7}}$ $1/(5+sqrt7)$ ?

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Answer 1 · 2016-04-05T17:29:59

$= \frac{5 - \sqrt{7}}{18}$ $= (5 - sqrt7) / 18$

Explanation:

$\frac{1}{5 + \sqrt{7}}$ $1 / ( 5 +sqrt7$

We rationalise the expression by multiplying it by the conjugate of the denominator. $(5 - \sqrt{7})$ $color(blue)( ( 5 - sqrt7)$

$= \frac{1 \cdot (5 - \sqrt{7})}{(5 + \sqrt{7}) \cdot (5 - \sqrt{7})}$ $= (1 * color(blue)(( 5 - sqrt7))) / (( 5 +sqrt7) * color(blue)( ( 5 - sqrt7))$

$= \frac{1 \cdot (5) + 1 \cdot (- \sqrt{7})}{(5 + \sqrt{7}) \cdot (5 - \sqrt{7})}$ $= (1 * color(blue)(( 5)) + 1 * color(blue)(( - sqrt7))) / (( 5 +sqrt7) * color(blue)( ( 5 - sqrt7))$

$= \frac{5 - \sqrt{7}}{(5 + \sqrt{7}) \cdot (5 - \sqrt{7})}$ $= (5 - sqrt7) / (( 5 +sqrt7) * color(blue)( ( 5 - sqrt7))$

Applying property:
$(a + b) (a - b) = a^{2} - b^{2}$ $color(blue)((a+b)(a-b) = a^2 - b^2$ to the denominator.

$= \frac{5 - \sqrt{7}}{(5^{2} - {(\sqrt{7})}^{2})}$ $= (5 - sqrt7) / (( 5 ^2 - (sqrt7)^2)$

$= \frac{5 - \sqrt{7}}{25 - 7}$ $= (5 - sqrt7) / (25 - 7)$

$= \frac{5 - \sqrt{7}}{18}$ $= (5 - sqrt7) / 18$

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