remove the irrationality in the denominator
(i) 1/1+√2+√3
1/(1+√2+√3) * 1/(1-(√2+√3))/(1-(√2+√3))
= (1-√2-√3)/(1 - (√2+√3)^2)
= (1-√2-√3)/(-4-2√6)
= (1-√2-√3)/(-2(2+√6))
= (1-√2-√3)/(-2(2+√6)) * (2-√6)/(2-√6)
= (1-√2-√3)(2-√6)/-2 * 1/(4-6)
= (2+√2-√6)/4
Right
Good
Right
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To remove the irrationality in the denominator, we need to rationalize it.
In the given expression, the denominator is 1 + √2 + √3. To rationalize this, we can multiply the numerator and denominator by the conjugate of the denominator.
The conjugate of 1 + √2 + √3 is 1 - √2 - √3.
So, multiplying the numerator and denominator by the conjugate, we get:
[(1/1+√2+√3) * (1 - √2 - √3) / (1 - √2 - √3)]
Now, we simplify the expression:
[(1 * 1 - √2 - √3) / (1 - √2 - √3)]
= (1 - √2 - √3) / (1 - √2 - √3)
= 1
Therefore, the simplified expression without the irrationality in the denominator is 1.