1)Identify the conjugate for 6 – 7√3 and explain your reasoning. 2)Multiply the radical expression 6 – 7√3 and its conjugate and write in simplified form. Show all your steps
3) Explain why we always get an integer when multiplying a radical expression by its conjugate? Provide your own appropriate example with supporting work that matches your explanation..

1) The conjugate for is .

2) -111

3) It is because number and its conjugate have the same numbers but their second components are opposite to each other.

Step-by-step explanation:

1) A radical number of the form , , where the first component is a non-radical real number which acts as a "pivot" and the second component is the radical component, which generates a "displacement" from pivot.

The conjugate of is a real number ,  , which that is "pivot" plus "displacement" in the direction opposite to that of . We proceed to complement this explanation with the image attached below.

Then, the conjugate for is .

2) We proceed to performed all the need algebraic operation until result is found:

1) Given

2) Distributive property

3) Definition of subtraction/

4) Distributive, associative and commutative properties

5) Definition of multiplication//Definition of square root

6) Commutative, associative and distributive properties/

7) Definition of subtraction/Existence of additive inverse

8)

9) Modulative property/Result

3) It is because number and its conjugate have the same numbers but their second components are opposite to each other.

the quotient is 4x^2 - 27x + 167 with a remainder of -996

step-by-step explanation:

in synthetic division, use -6 as the divisor and use the coefficients {4, -3, 5, 6}:

-6     )   4     -3     5     6

                -24   162 -1002

          4   -27   167   -996

thus, the quotient is 4x^2 - 27x + 167 with a remainder of -996.

note:   your "4x°-3x+5x+6" is ambiguous / incomplete.   i had to assume that you meant '4x^3 - 3x^2 + 5x + 6."