Answer:
1) The conjugate for [tex]6-7\cdot \sqrt{3}[/tex] is [tex]6+7\cdot \sqrt{3}[/tex].
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 [tex]r = a + b\sqrt{c}[/tex], [tex]\forall \,a,b,c\in \mathbb{R}[/tex], 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 [tex]r[/tex] is a real number [tex]s = a - b\sqrt{c}[/tex], [tex]\forall \,a,b,c\in \mathbb{R}[/tex], which that is "pivot" plus "displacement" in the direction opposite to that of [tex]r[/tex]. We proceed to complement this explanation with the image attached below.
Then, the conjugate for [tex]6-7\cdot \sqrt{3}[/tex] is [tex]6+7\cdot \sqrt{3}[/tex].
2) We proceed to performed all the need algebraic operation until result is found:
1) [tex](6+7\sqrt{3})\cdot (6-7\sqrt{3})[/tex] Given
2) [tex]6\cdot (6-7\sqrt{3})+7\sqrt{3}\cdot (6-7\sqrt{3})[/tex] Distributive property
3) [tex]6\cdot [6+(-7)\cdot \sqrt{3}]+7\cdot \sqrt{3}\cdot [6+(-7)\cdot \sqrt{3}][/tex] Definition of subtraction/ [tex]-a\cdot b = (-a)\cdot b[/tex]
4) [tex]6\cdot 6 +[6\cdot (-7)]\cdot \sqrt{3}+(6\cdot 7)\cdot \sqrt{3}+[7\cdot (-7)]\cdot (\sqrt{3}\cdot \sqrt{3})[/tex] Distributive, associative and commutative properties
5) [tex]36 +(-42)\cdot \sqrt{3}+42\cdot \sqrt{3}+(-49)\cdot 3[/tex] Definition of multiplication/[tex]-a\cdot b = (-a)\cdot b[/tex]/Definition of square root
6) [tex][36 +(-147)]+\sqrt{3}\cdot [42+(-42)][/tex] Commutative, associative and distributive properties/[tex]-a\cdot b = (-a)\cdot b[/tex]
7) [tex]-111+\sqrt{3}\cdot 0[/tex] Definition of subtraction/Existence of additive inverse
8) [tex]-111+0[/tex] [tex]a\cdot 0 = 0[/tex]
9) [tex]-111[/tex] Modulative property/Result
3) It is because number and its conjugate have the same numbers but their second components are opposite to each other.
