Question

Multiply, and then simplify the problem. Assume that all variables represent positive real numbers.
[(√2+√3)-√6] [(√2+√3)+√6]

Answer 1

variable?

Answer 2

use the difference of squares identity :D
(identity: \(a^2-b^2=(a+b)(a-b)\))

Answer 3

I should use FOIL on the binomials, and then what am I suppose to do with the -√6 and √6?

Answer 4

no you do this in your mind: a=\(\sqrt2+\sqrt3\), b=\(\sqrt6\)

Answer 5

\([(\sqrt2+\sqrt3)-\sqrt6][(\sqrt2+\sqrt3)+\sqrt6]\)
=\((\sqrt2+\sqrt3)^2-\sqrt6^2\)

Answer 6

Is that how it would be simplified?

Answer 7

this is just the first step :D
(do you get the first step)

Answer 8

I see how you got that, but how does it end up being: 2√6-1 in the answer key?

Answer 9

why would it not? :)

Answer 10

Do I use foil to get that 2? I know that √2*√2=2. I thought that was how they got that part, but I don't see where the -1 comes from.

Answer 11

don't let the answer key affect you

Answer 12

\([(\sqrt2+\sqrt3)-\sqrt6][(\sqrt2+\sqrt3)+\sqrt6]\)
=\((\sqrt2+\sqrt3)^2-\sqrt6^2\)
=\((\sqrt2^2+2\sqrt2\sqrt3+\sqrt3^2)-6\)

Answer 13

do you actually remember the identities

Answer 14

Let me type the idens all out

Answer 15

\(1.\) \((a+b)^2=a^2+2ab+b^2\)
\(2.\) \((a-b)^2=a^2-2ab+b^2\)
\(3.\) \(a^2-b^2=(a+b)(a-b)\)
\(4.\) \(a^3+b^3=(a+b)(a^2-ab+b^2)\)
\(5.\) \(a^3-b^3=(a-b)(a^2+ab+b^2)\)

Answer 16

so now we have:
\([(\sqrt2+\sqrt3)-\sqrt6][(\sqrt2+\sqrt3)+\sqrt6]\)
=\((\sqrt2+\sqrt3)^2-\sqrt6^2\)
=\((\sqrt2^2+2\sqrt2\sqrt3+\sqrt3^2)-6\)

could you continue? :)

Answer 17

I wouldn't know what to do with that. Am I suppose to somehow simplify that further? Or do I just do something with the -6 first?

Answer 18

simplify that further :)

Answer 19

How do I simplify it further? Is it: (2√2^3+√3^3)-6?

Answer 20

nope

Answer 21

what is the definition of sqrt?

Answer 22

ok let's do it terms by term

Answer 23

look at what we got here:
\((\sqrt2^2+2\sqrt2\sqrt3+\sqrt3^2)-6\)

The first term \(\sqrt2^2\) is equivalent to?

Answer 24

2

Answer 25

and the second term \(2\sqrt2\sqrt3\) is?

Answer 26

2√6

Answer 27

and the third?

Answer 28

3

Answer 29

and the whole thing?

Answer 30

I see it now! It's: 5+2√6-6, which turns into: 2√6-1. That's great, thank you!

Answer 31

no problem :)