Question

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

1) The conjugate for
`6-7\cdot √(3)` is
`6+7\cdot √(3)`.

2) -111

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

Explanation:

1) A radical number of the form
`r = a + b√(c)`,
`\forall \,a,b,c\in \mathbb{R}`, 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
`r` is a real number
`s = a - b√(c)`,
`\forall \,a,b,c\in \mathbb{R}`, which that is "pivot" plus "displacement" in the direction opposite to that of
`r`. We proceed to complement this explanation with the image attached below.

Then, the conjugate for
`6-7\cdot √(3)` is
`6+7\cdot √(3)`.

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

1)
`(6+7√(3))\cdot (6-7√(3))` Given

2)
`6\cdot (6-7√(3))+7√(3)\cdot (6-7√(3))` Distributive property

3)
`6\cdot [6+(-7)\cdot √(3)]+7\cdot √(3)\cdot [6+(-7)\cdot √(3)]` Definition of subtraction/
`-a\cdot b = (-a)\cdot b`

4)
`6\cdot 6 +[6\cdot (-7)]\cdot √(3)+(6\cdot 7)\cdot √(3)+[7\cdot (-7)]\cdot (√(3)\cdot √(3))` Distributive, associative and commutative properties

5)
`36 +(-42)\cdot √(3)+42\cdot √(3)+(-49)\cdot 3` Definition of multiplication/
`-a\cdot b = (-a)\cdot b`/Definition of square root

6)
`[36 +(-147)]+√(3)\cdot [42+(-42)]` Commutative, associative and distributive properties/
`-a\cdot b = (-a)\cdot b`

7)
`-111+√(3)\cdot 0` Definition of subtraction/Existence of additive inverse

8)
`-111+0`
`a\cdot 0 = 0`

9)
`-111` Modulative property/Result

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