Question

Rationalize the denominator and simplify.
https://media.glynlyon.com/g_alg02_ccss_2013/7/q11166.gif

Answer 1

Use the idea of the conjugate pair! (aka Difference of Squares)

\(a^{2} - b^{2} = (a+b)(a-b)\)

Answer 2

my answer is 20+3 sqrt3 but I know I'm wrong

Answer 3

@tkhunny

Answer 4

If you know you are wrong, why is that your answer? That makes no sense at all.

Did you do this? \(\dfrac{5}{4-\sqrt{3}}\cdot\dfrac{4+\sqrt{3}}{4+\sqrt{3}}\)

Answer 5

It's my answer to let you know that I tried. Yes, I did that.

Answer 6

Okay, then what did you get for Numerator and Denominator separately?

Answer 7

20+3 sqrt /16- sqrt9

Answer 8

Numerator should be \(20+5\sqrt{3}\). Do you see where you wandered off?

Denominator is perfect. Now, simplify it.

Notation needs a little work, too. These are NOT the same:

You Wrote: 20+3 sqrt /16- sqrt9
You Meant: (20+3 sqrt) /(16- sqrt9)

You need to know why those are different.

Answer 9

Where did the 5 come from? Why are those different?

Answer 10

That is a little disappointing. I blame your teachers or curriculum, not you.

The 5 came from the Distributive Property.

Multiplying Numerators: \(5\cdot (4+\sqrt{3}) = 5\cdot 4 + 5\cdot \sqrt{3} = 20+5\sqrt{3}\)

They are different because of the Order of Operations:

You Wrote: 20+3 sqrt /16- sqrt9 = \(20 + \dfrac{\sqrt{3}}{16} - \sqrt{9}\)

You Meant: (20+3 sqrt) /(16- sqrt9) = \(\dfrac{20+\sqrt{3}}{16-\sqrt{9}}\)

Now do you see the difference? Writing things in a line is not quite the same as writing things in a text book or on the board.

Answer 11

Thanks. Yes. So whats the next step?