What is 4/(5+sqrt(2)) ?

The solution to 4/(5+sqrt(2)) is (20-4sqrt(2))/(23)

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simplify 4/(5+sqrt(2))

simplify

\frac{4}{5 + 2}

\frac{20 - 4 2}{23}

Solution steps

\frac{4}{5 + 2}

Multiply by the conjugate

\frac{5 - 2}{5 - 2}

= \frac{4 ( 5 - 2 )}{( 5 + 2 ) ( 5 - 2 )}

Expand

4 (5 - 2) : 20 - 42

Expand

(5 + 2) (5 - 2) : 23

= \frac{20 - 4 2}{23}

\frac{22}{- 4 + 5}

100 - 3

4 (x + 2 y) - (x - y)

4 (2 + 5 i) - (3 - 4 i)

\frac{20}{3} \times 6