Question #b98d8

1 Answer
Jan 15, 2017

Answer:

Here's how you can do that.

Explanation:

Your starting expression looks like this

#12/(4 - sqrt(3))#

Now, the first thing you need to do here is to rationalize the denominator, which is another way of saying that we need to eliminate that radical term from the denominator.

To do that, you can multiply it by its conjugate. To get the conjugate of this expression

#4 - sqrt(3)#

you simply change the sign that you have in the middle of the two terms, i.e. you need to change the sign of the second term.

In this case, the minus sign that precedes #sqrt(3)# will become a plus sign

#4 - sqrt(3) " "stackrel(color(white)(acolor(blue)("change the sign of the second term")aaa))(->) 4 color(red)(+)sqrt(3)#

Now, you can get rid of the radical term by multiplying the original denominator by its conjugate

#(4 -sqrt(3)) * (4 + sqrt(3))#

Keep in mind that you have

#color(blue)(ul(color(black)(a^2 - b^2 = (a-b)(a+b))))#

which means that you can write

#(4 -sqrt(3)) * (4 + sqrt(3)) = 4^2 - (sqrt(3))^2#

#=16 - 3#

#=13#

In order to be able to multiply the denominator by its conjugate, you must multiply the original expression by #1 = (4 + sqrt(3))/(4 + sqrt(3))#.

This will give you

#12/(4 - sqrt(3)) * (4 + sqrt(3))/(4 + sqrt(3)) = (12 * (4 + sqrt(3)))/((4 - sqrt(3))(4 + sqrt(3))#

You can thus say that your original expression is equivalent to

#12/(4 - sqrt(3)) = 12/13 * (4 + sqrt(3))#