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..

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.

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