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7+3√5/3+√5 - 7-3√5/3-√5 = a + b√5

find a and b
I'm currently stuck at a certain part, here is a picture of the question

7+3√5/3+√5 - 7-3√5/3-√5 = a + b√5 find a and b I'm currently stuck at a certain part-example-1
User Helen
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1 Answer

7 votes


\begin{gathered} \\(7 + 3 √(5) )/(3 + √(5) ) - (7 - 3 √(5) )/(3 - √(5) ) = a + √(5) b \\ \end{gathered}


\begin{gathered} \\ (( \: 7 + 3 √(5) \: \: ( 3 - √(5)) \: \: - 7 - 3 √(5) \: \: ( 3 + √(5)) \: )/((3 + √(5)) \: \:(3 + √(5)) ) = a + √(5) \: b\\ \end{gathered}


\begin{gathered} \\ (( \: 21 - 7 √(5) \: + 9 √(5) - 15) \: \: - ( \: 21 + 7 √(5) \: - 9 √(5) + 15)\: )/((3 + √(5)) \: \:(3 + √(5)) ) = a + √(5) \: b\\ \end{gathered}


\begin{gathered} \\ (( \: 6 + 2 √(5) ) \: \: - ( \: 6 - 2 √(5) )\: )/((3 + √(5)) \: \:(3 + √(5)) ) = a + √(5) \: b\\ \end{gathered}


\begin{gathered} \\ (\: 6 + 2 √(5) \: \: - \: \: 6 - 2 √(5) \: )/((3 + √(5)) \: \:(3 + √(5)) ) = a + √(5) \: b\\ \end{gathered}


\begin{gathered} \\ \frac{\: 4 √(5) \: \: }{3 {}^(2) - {√(5) }^(2) } = a + √(5) \: b\\ \end{gathered}


\begin{gathered} \\ (\: 4 √(5) \: \: )/( \: \: \: \: 9 - 5 \: \: \: ) = a + √(5) \: b\\ \end{gathered}


\begin{gathered} \\ (\: 4 √(5) \: \: )/( \: \: \: \: 4 \: \: \: ) = a + √(5) \: b\\ \end{gathered}


\begin{gathered} \\ \: √(5) = a + √(5) \: b\\ \end{gathered}

we can also write it as ;


\begin{gathered} \\ \: 0 + √(5) = a + √(5) \: b\\ \end{gathered}

★ Henceforth, the value of a and b are :

→ a = 0

→ b = 1

User Ron Newcomb
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