To multiply each expression by its conjugate, you need to remember that the conjugate of a number is obtained by changing the sign between two terms. For example, the conjugate of a + b is a - b.
Let's go through each expression and multiply it by its conjugate:
A. Square root of (13) - 2:
Take the expression (β13 - 2) and multiply it by its conjugate, (β13 + 2):
(β13 - 2)(β13 + 2)
= (β13)^2 - (2)^2 [using the difference of squares formula]
= 13 - 4
= 9
So the answer for expression A is 9.
B. 2 + 3β5:
Take the expression (2 + 3β5) and multiply it by its conjugate, which is (2 - 3β5):
(2 + 3β5)(2 - 3β5)
= (2)^2 - (3β5)^2 [using the difference of squares formula]
= 4 - 9(5)
= 4 - 45
= -41
So the answer for expression B is -41.
C. Square root(3) minus Square root(7):
Take the expression (β3 - β7) and multiply it by its conjugate, which is (β3 + β7):
(β3 - β7)(β3 + β7)
= (β3)^2 - (β7)^2 [using the difference of squares formula]
= 3 - 7
= -4
So the answer for expression C is -4.
To solve these types of problems, multiply each expression by its conjugate and simplify using the difference of squares formula if applicable.