3-square root 2/ 1+square root 2
This is not trig.
(3 - sqrt2)/(1 + sqrt2)
= (3-sqrt2)(1-sqrt2)/[(1-sqrt2)(1+sqrt2)]
= [3 - 4sqrt2 +2]/(1-2)
= (5 -4sqrt2)/-1
= 4sqrt2 -5
To simplify the expression (3√2) / (1 + √2), we need to rationalize the denominator.
Rationalizing means getting rid of any radical expressions (square roots) in the denominator. To do this, we can multiply the numerator and denominator by the conjugate of the denominator.
The conjugate of a binomial expression (a ± b) is obtained by changing the sign between the terms, i.e., (a ∓ b).
In this case, the conjugate of (1 + √2) is (1 - √2).
Now, let's multiply the numerator and denominator by the conjugate:
[(3√2) / (1 + √2)] * [(1 - √2) / (1 - √2)]
Multiplying the numerators and denominators separately:
Numerator: (3√2) * (1 - √2) = 3√2 - 3(√2)^2 = 3√2 - 3(2) = 3√2 - 6
Denominator: (1 + √2) * (1 - √2) = 1 - (√2)^2 = 1 - 2 = -1
Therefore, the simplified expression is: (3√2 - 6) / -1
However, it is common practice to multiply both the numerator and denominator by -1 to change the sign of the expression. Doing this would yield:
-(3√2 - 6) / 1
Simplifying further, we have:
-(3√2) + 6
So, the simplified form of the given expression is -(3√2) + 6.