by rationalizing the denominator simplify 1+√5/1-√5
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You obviously mean
(1+√5)/(1-√5) or else there would be no problem
= (1+√5)/(1-√5) * (1+√5)/(1+√5)
= (1 + 2√5 + 5) / (1 - 5)
= (6 + 2√5)/-4
= (-3 - √5)/2
To rationalize the denominator of the given expression, we need to multiply both the numerator and denominator by the conjugate of the denominator. In this case, the conjugate of 1-√5 is 1+√5.
So, we have:
(1+√5) / (1-√5)
Now, let's multiply the numerator and denominator by the conjugate (1+√5):
[(1+√5) * (1+√5)] / [(1-√5) * (1+√5)]
Expanding the numerator and denominator:
[(1+√5)(1+√5)] / [(1^2 - (√5)^2)]
Simplifying further:
[1 + √5 + √5 + 5] / [1 - 5]
Combining like terms:
[6 + 2√5] / (-4)
Finally, we can simplify further by dividing both the numerator and denominator by their greatest common divisor, which is 2:
[3 + √5] / (-2)
Therefore, the simplified expression, after rationalizing the denominator, is (3 + √5) / (-2).
To simplify the expression and rationalize the denominator, we need to get rid of the square root in the denominator by multiplying both the numerator and denominator by the conjugate of the denominator.
The conjugate of 1 - √5 is 1 + √5.
So, we can multiply both the numerator and denominator by 1 + √5:
(1 + √5)/(1 - √5) * (1 + √5)/(1 + √5)
Expanding this expression, we get:
(1 + √5)(1 + √5) / ((1 - √5)(1 + √5))
Now, let's simplify each portion separately:
Numerator: (1 + √5)(1 + √5) = 1 + √5 + √5 + 5 = 6 + 2√5
Denominator: (1 - √5)(1 + √5) = 1 + √5 - √5 - 5 = -4
Putting it all together, the simplified expression is:
(6 + 2√5) / (-4)
You can further simplify this by dividing both the numerator and the denominator by their greatest common factor, which is 2:
(6/2 + 2√5/2) / (-4/2)
Simplifying further:
3 + √5 / -2
So, 1 + √5 / 1 - √5, after rationalizing the denominator, simplifies to:
3 + √5 / -2