Rationalize the denominator of the expression:
sqrt2+1/sqrt-1
(√2 + 1)/i
(√2 + 1)i/i^2
-(√2 + 1)i
To rationalize the denominator of the expression, we want to eliminate any radicals in the denominator. In this case, we have a square root in the denominator, which is more commonly referred to as an imaginary number. To rationalize the denominator, we need to multiply both the numerator and denominator by the conjugate of the denominator.
The conjugate of a square root expression with a binomial denominator is obtained by changing the sign between the terms in the binomial. In this case, the conjugate of sqrt(-1) is sqrt(-1) as well.
So, to rationalize the denominator, we multiply the numerator and denominator by the conjugate of the denominator, which is sqrt(-1).
(sqrt(2) + 1/sqrt(-1)) * (sqrt(-1)/sqrt(-1))
When multiplying the numerator and denominator, we use the property of the conjugate, which states that (a + b)(a - b) = a^2 - b^2.
Therefore, applying this property, we get:
(sqrt(2)*sqrt(-1) + 1*sqrt(-1))/(sqrt(-1)*sqrt(-1))
Simplifying this expression, we have:
sqrt(-2) - sqrt(-1)/(-1)
Since sqrt(-1) is equal to "i", we can rewrite the expression as:
sqrt(-2) - i/(-1)
Multiplying the numerator and denominator by -1, we get:
sqrt(-2) + i
Therefore, the rationalized expression is sqrt(-2) + i.