Question

please give me correct answer SIMPLIFY​​

Answer 1

Question:-

Simplify the following:-

(7 + 3√5)/(3 + √5) – (7 - 3√5)/(3 - √5)

Solution:-

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

Let us solve the parts one by one.

For Left part:-

$= \sf{\dfrac{7 + 3\sqrt{5}}{3 + \sqrt{5}}} </p><p>$

By Rationalizing the denominator:-

$= \sf{\dfrac{7 + 3\sqrt{5}}{3 + \sqrt{5}} \times \dfrac{3 - \sqrt{5}}{3 - \sqrt{3}}}$

$= \sf{\dfrac{(7 + 3\sqrt{5})(3 - \sqrt{5})}{(3 + \sqrt{5})(3 - \sqrt{5})}}$

We know,

• (a + b)(a - b) = a² - b²

$= \sf{\dfrac{7(3 - \sqrt{5}) + 3\sqrt{5}(3 - \sqrt{5})}{(3)^2 - (\sqrt{5})^2}}$

$= \sf{\dfrac{21 - 7\sqrt{5} + 9\sqrt{5} - 15}{9 - 5}}$

$= \sf{\dfrac{6 + 2\sqrt{5}}{4}}$

$= \sf{\dfrac{2(3 + \sqrt{5}}{4}}$

$= \sf{\dfrac{3 + \sqrt{5}}{2}}$

For Left part:-

$\sf{\dfrac{7 - 3\sqrt{5}}{3 - \sqrt{5}}}$

Rationalising the denominator,

$= \sf{\dfrac{7 - 3\sqrt{5}}{3 - \sqrt{5}} \times \dfrac{3 + \sqrt{5}}{3 + \sqrt{3}}}$

$= \sf{\dfrac{(7 - 3\sqrt{5})( 3 + \sqrt{5})}{(3 - \sqrt{5})(3 + \sqrt{5})}}$

We know,

• (a + b)(a - b) = a² - b²

$= \sf{\dfrac{7(3 + \sqrt{5}) - 3\sqrt{5}(3 + \sqrt{5})}{(3)^2 - (\sqrt{5})^2}}$

$= \sf{\dfrac{21 + 7\sqrt{5} - 9\sqrt{5} - 15}{9 - 5}}$

$= \sf{\dfrac{6 - 2\sqrt{5}}{4}}$

$= \sf{\dfrac{2(3 - \sqrt{5}}{4}}$

$= \sf{\dfrac{3 - \sqrt{5}}{2}}$

Therefore,

$\sf{\dfrac{7 + 3\sqrt{5}}{3 + \sqrt{5}} = \dfrac{3 + \sqrt{5}}{2}}$

And,

$\sf{\dfrac{7 - 3\sqrt{5}}{3 - \sqrt{5}} = \dfrac{3 - \sqrt{5}}{2}}$

Hence,

$The \: value \: of \sf{\dfrac{7 + 3\sqrt{5}}{3 + \sqrt{5}} - \dfrac{7 - 3\sqrt{5}}{3 - \sqrt{5}}}$

is as follows:-

$= \sf{\dfrac{3 + \sqrt{5}}{2} - \dfrac{3 - \sqrt{5}}{2}}$

$= \sf{\dfrac{3 + \sqrt{5} - (3 - \sqrt{5})}{2}}$

$= \sf{\dfrac{3 + \sqrt{5} - 3 + \sqrt{5}}{2}}$

$= \sf{\dfrac{2\sqrt{5}}{2}}$

$= √5$

∴ The answer is √5.

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