Question

Find the value of 'a' and 'b' in
$\frac{7 + 3 \sqrt{5} }{3 + \sqrt{5} } - \frac{7 - 3 \sqrt{5} }{3 - \sqrt{5} } = a + b \sqrt{5}$
plz solve correctly and do not spam​

Answer 1

Answer:

${ \large{ \underline{ \sf{Given}}}}$

• $\sf{ \frac{7 + 3 \sqrt{5} }{3 + \sqrt{5} } - \frac{7 - 3 \sqrt{5} }{3 - \sqrt{5} } = a + b \sqrt{5}}$

${ \large{ \underline{ \sf{To \: Find}}}} \:$

• Value of a and b

${ \large{ \underline{ \sf{Solution }}}}$

• First let us rationalize the denominator and solve it at last we will get the value of a and b. Let's start our solution.

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$\dashrightarrow{ \sf{ \frac{7 + 3 \sqrt{5} }{3 + \sqrt{5} } - \frac{7 - 3 \sqrt{5} }{3 - \sqrt{5} } }} \\ \\ \\ \dashrightarrow{ \sf{ \frac{7+ 3\sqrt{5} }{3+ \sqrt{5} } \times \frac{3 - \sqrt{5} }{3- \sqrt{5} } - \bigg( \frac{7-3 \sqrt{5} }{3- \sqrt{5} } × \frac{3+ \sqrt{5} }{3+ \sqrt{5} }\bigg) }} \\ \\ \\ \dashrightarrow{ \sf{ \frac{3- \sqrt{5} (7 + 3 \sqrt{5} )}{ {3}^{2} - {( \sqrt{5} )}^{2} } - \bigg( \frac{3+ \sqrt{5} (7-3 \sqrt{5} )}{ {3}^{2} - { (\sqrt{5} )}^{2} } }} \\ \\ \\ \dashrightarrow{ \sf{ \frac{21 + 9 \sqrt{5} - 7 \sqrt{5} - 15}{9 - 5} - \bigg( \frac{21 + 7 \sqrt{5} - 9 \sqrt{5} - 15}{9 - 5} \bigg)}} \\ \\ \\ \dashrightarrow{ \sf{ \frac{21 + 9 \sqrt{5} - 7 \sqrt{5} - 15 - 21 - 7 \sqrt{5} + 9 \sqrt{5} + 15 }{ 4 } }} \\ \\ \\ \dashrightarrow{ \sf{ \frac{9 \sqrt{5} + 9 \sqrt{5} - 7 \sqrt{5} - 7 \sqrt{5} }{4} }} \\ \\ \\ \dashrightarrow{ \sf{ \frac{18 \sqrt{5} - 14 \sqrt{5} }{4} }} \\ \\ \\ \dashrightarrow{ \sf{ \frac{4 \sqrt{5} }{4} }} \\ \\ \\ \dashrightarrow{ \sf{ \sqrt{5} }} \\ \\ \\ { \sf{Comparing \: with \: a + b \sqrt{5} }} \\ \\ \\ \sf{We \: get, } \\ \\ \\ \sf{a = 0 ,b = 1}$

Therefore,

• Values of a and b are 0, 1

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