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$Express (3-5sqrt5)/(3+2sqrt5) in the form (asqrt5-b) where a and b are simple fractions.$

Express $\frac{3-5 \sqrt{5}}{3+2 \sqrt{5}}$ in the form $(a \sqrt{5}-b)$ where $a$ and $b$ are simple fractions.

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Double integration by reversing the order

Full solution

Q. Express

\frac{3-5 \sqrt{5}}{3+2 \sqrt{5}}

in the form

(a \sqrt{5}-b)

where

a

and

b

are simple fractions.

Multiply by Conjugate: To simplify the expression $(3-5\sqrt{5})/(3+2\sqrt{5})$ into the form $(a\sqrt{5}-b)$ , we can multiply the numerator and the denominator by the conjugate of the denominator to eliminate the square root in the denominator. $\newline$ The conjugate of $(3+2\sqrt{5})$ is $(3-2\sqrt{5})$ . $\newline$ We will multiply both the numerator and the denominator by this conjugate.
Expand Numerator: Now, let's perform the multiplication: $\newline$ Numerator: $(3-5\sqrt{5}) \times (3-2\sqrt{5})$ $\newline$ Denominator: $(3+2\sqrt{5}) \times (3-2\sqrt{5})$
Expand Denominator: Let's first expand the numerator: $\newline$ $(3-5\sqrt{5}) \times (3-2\sqrt{5}) = 3\times3 - 3\times2\sqrt{5} - 5\sqrt{5}\times3 + 5\sqrt{5}\times2\sqrt{5}$ $\newline$ $= 9 - 6\sqrt{5} - 15\sqrt{5} + 10\times5$ $\newline$ $= 9 - 21\sqrt{5} + 50$ $\newline$ $= 59 - 21\sqrt{5}$
Expand Denominator: Let's first expand the numerator: $\newline$ $(3-5\sqrt{5}) \times (3-2\sqrt{5}) = 3\times3 - 3\times2\sqrt{5} - 5\sqrt{5}\times3 + 5\sqrt{5}\times2\sqrt{5}$ $\newline$ $= 9 - 6\sqrt{5} - 15\sqrt{5} + 10\times5$ $\newline$ $= 9 - 21\sqrt{5} + 50$ $\newline$ $= 59 - 21\sqrt{5}$ Now, let's expand the denominator: $\newline$ $(3+2\sqrt{5}) \times (3-2\sqrt{5}) = 3\times3 - 3\times2\sqrt{5} + 2\sqrt{5}\times3 - 2\sqrt{5}\times2\sqrt{5}$ $\newline$ $= 9 - 6\sqrt{5} + 6\sqrt{5} - 4\times5$ $\newline$ $= 9 - 20$ $\newline$ $= -11$

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