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If a =2 + root 3 then find 1/a |
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Answer» Answer: Here is the answer you were looking for: \begin{GATHERED}a = 2 + \SQRT{3} \\ \\ \frac{1}{a} = \frac{1}{2 + \sqrt{3} } \\ \end{gathered} a=2+ 3
a 1
= 2+ 3
1
On rationalizing the denominator we GET, \begin{gathered} \frac{1}{a} = \frac{1}{2 + \sqrt{3} } \times \frac{2 - \sqrt{3} }{2 - \sqrt{3} } \\ \end{gathered} a 1
= 2+ 3
1
× 2− 3
2− 3
USING the identity : (x + y)(x - y) = {x}^{2} - {y}^{2}(x+y)(x−y)=x 2 −y 2
\begin{gathered} \frac{1}{a} = \frac{2 - \sqrt{3} }{ {(2)}^{2} - {( \sqrt{3} )}^{2} } \\ \\ \frac{1}{a} = \frac{2 - \sqrt{3} }{4 - 3} \\ \\ \frac{1}{a} = 2 - \sqrt{3} \\ \\ a - \frac{1}{a} \end{gathered} a 1
= (2) 2 −( 3
) 2
2− 3
a 1
= 4−3 2− 3
a 1
=2− 3
a− a 1
Putting the values, \begin{gathered}a - \frac{1}{a} = (2 + \sqrt{3} ) - (2 - \sqrt{3} ) \\ \\ a - \frac{1}{a} = 2 + \sqrt{3} - 2 + \sqrt{3} \\ \\ a - \frac{1}{a} = \sqrt{3} + \sqrt{3} \\ \\ a - \frac{1}{a} = 2 \sqrt{3} \end{gathered} a− a 1
=(2+ 3
)−(2− 3
) a− a 1
=2+ 3
−2+ 3
a− a 1
= 3
+ 3
a− a 1
=2 3
Hope this HELPS!!! |
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