How do you rationalize the denominator and simplify $(5+ sqrt 5)/(8- sqrt 5)$ ?

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Answer 1 · 2015-04-10T17:10:16

Multiply the numerator an denominator by the conjugate of the denominator, then look for simplification factors

$(5+sqrt(5))/(8-sqrt(5))$

$=(5+sqrt(5))/(8-sqrt(5)) * (8+sqrt(5))/(8+sqrt(5))$

$= (40 +13sqrt(5) +5)/(64-5)$

$= (45+13sqrt(5))/59$

Answer 2 · 2015-04-10T17:14:27

Multiply the fraction by $1$ in the form $(8+sqrt5)/(8+sqrt5)$

$((5+sqrt5))/((8-sqrt5)) ((8+sqrt5))/((8+sqrt5))=(40+5sqrt5+8sqrt5+5)/(64-5)$

$=(45 + 13 sqrt5)/59$ .

This works because of the product: $(a-b)(a+b)=a^2-b^2$ .

If one or both ob $a$ , $b$ have square roots, then the product doesn't: for example: $(a-sqrtc)(a+sqrtc)=a^2 - (sqrtc)^2 = a^2-c$ .

How do you rationalize the denominator and simplify (5+ sqrt 5)/(8- sqrt 5)(5+ sqrt 5)/(8- sqrt 5)?