1.

7+3​

Answer»

Answer:

\large{\red{\boxed{\sf \frac{7 + 3 \sqrt{5} }{3 + \sqrt{5}} - \frac{7 - 3 \sqrt{5} }{3 - \sqrt{5} } = \sqrt{5}}}}

Step-by-step explanation:

GIVEN that ;

\dfrac{7 + 3 \sqrt{5} }{3 + \sqrt{5} } - \dfrac{7 - 3 \sqrt{5} }{3 - \sqrt{5} }

LET a = \sf \dfrac{7 + 3\sqrt{5}}{3 + \sqrt{5}} and,

B = \sf \dfrac{7 - 3\sqrt{5}}{3 - \sqrt{5}}

We need to FIND, (a - b) -

CONSIDER a;

\frac{7 + 3 \sqrt{5} }{3 + \sqrt{5} } \\ \\ \implies \frac{7 + 3 \sqrt{5} }{3 + \sqrt{5} } \times \frac{3 - \sqrt{5} }{3 - \sqrt{5} } \\ \\ \implies \frac{(7 + 3 \sqrt{5} )(3 - \sqrt{5} )}{(3) {}^{2} - ( \sqrt{5}) {}^{2} } \\ \\ \implies \frac{35 + 9 \sqrt{5} - 7 \sqrt{5} - 15 }{9 - 5} \\ \\ \implies \frac{20 + 2 \sqrt{5} }{4} \\ \\ \implies \frac{2(10 + \sqrt{5}) }{4}\\ \\ \implies \frac{10 + \sqrt{5} }{2}

Consider b;

\frac{7 - 3 \sqrt{5} }{3 - \sqrt{5} } \\ \\ \implies \frac{7 - 3 \sqrt{5} }{3 - \sqrt{5} } \times \frac{3 + \sqrt{5} }{3 + \sqrt{5} } \\ \\ \implies \frac{(7 - 3 \sqrt{5})(3 + \sqrt{5})}{(3) {}^{2} - ( \sqrt{5}) {}^{2} } \\ \\ \implies \frac{35 + 7 \sqrt{5} - 9 \sqrt{5} - 15}{9 - 5} \\ \\ \implies \frac{20 - 2 \sqrt{5} }{4} \\ \\ \implies \frac{2(10 - \sqrt{5}) }{4} \\ \\ \implies \frac{10 - \sqrt{5} }{2}

Now, find (a - b) -

\implies \frac{10 + \sqrt{5} }{2} - \frac{10 - \sqrt{5} }{2} \\ \\ \implies \frac{10 + \sqrt{5} - (10 - \sqrt{5}) }{2} \\ \\ \implies \frac{10 + \sqrt{5} - 10 + \sqrt{5} }{2} \\ \\ \implies \frac{2 \sqrt{5}}{ 2} \\ \\ \boxed{ \bf \therefore \: \frac{7 + 3 \sqrt{5} }{3 + \sqrt{5}} - \frac{7 - 3 \sqrt{5} }{3 - \sqrt{5} } = \sqrt{5} }


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