Which operations have rational or irrational result?

Step-by-step explanation:

Square roots are all irrational except a very few that are perfect squares.

The top one is irrational because sqrt(3) is irrational. That fact that it has a rational (3) added on does not matter. This is still irrational.

The second one is irrational for the same reason.

The third one is an odd result. The two factor binomials are both irrational, but when multiplied, give a rational result

(4 - sqrt(5)) * (4 + sqrt(5)) = 16 - sqrt(5)*sqrt(5) = 16 - 5 = 11

The fourth one is also a bit tricky. Focus on sqrt(6) for a second.

sqrt(6) = sqrt(2)*sqrt(3) Try this on your calculator. Since there is a sqrt(3) in both the numerator and the denominator they cancel. The Two sqrt(2)'s multiply together to give 2 which cancels with two in the numerator. The result is rational.  It is 2.

The fifth one gives 4^(1/2) = 2; 27^(1/3) = 3

2 + 3 = 5

See if you can figure out why the sixth one is irrational.

x is greater than 5

step-by-step explanation:

2x+7 is greater than 17

    -7                           -7

2x is greater than 10

divide by 2 each side

x is greater than 5

so far i think you have the right thing going for you just remember soh cah toh

step-by-step explanation:

sine over hypotenuse

cosine   equals adjacent

tangent over hypotenuse