If \(a = 9 - 4 \sqrt{5}\), then find the value of \(a - \frac{1}{a}\).
Answer (2)
The value of a − a 1 is − 8 5 , where a = 9 − 4 5 . This was calculated by finding a 1 using the conjugate of the denominator and then subtracting. Thus, the final answer is a − a 1 = − 8 5 .
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To find the value of a − a 1 given that a = 9 − 4 5 , we can follow these steps:
Substitute the given value :
We have a = 9 − 4 5 . To find a 1 , we need to calculate the reciprocal of a .
Find the reciprocal a 1 :
To find the reciprocal of a , rationalize the denominator:
a 1 = 9 − 4 5 1 . To rationalize, we'll multiply the numerator and denominator by the conjugate of the denominator, which is 9 + 4 5 :
9 − 4 5 1 ⋅ 9 + 4 5 9 + 4 5 = ( 9 − 4 5 ) ( 9 + 4 5 ) 9 + 4 5 .
Simplify the expression :
The denominator simplifies using the difference of squares formula:
( 9 − 4 5 ) ( 9 + 4 5 ) = 9 2 − ( 4 5 ) 2 = 81 − 80 = 1. Therefore: a 1 = 9 + 4 5 .
Calculate a − a 1 :
Now substitute a = 9 − 4 5 and a 1 = 9 + 4 5 into the expression:
a − a 1 = ( 9 − 4 5 ) − ( 9 + 4 5 ) . Simplify: a − a 1 = 9 − 4 5 − 9 − 4 5 = − 8 5 .
Therefore, the value of a − a 1 is − 8 5 .
