Question

Can someone please walk me through solving this problem? I have really been struggling with it.

Which choice is equivalent to the fraction below when x is an appropriate value? Hint: Rationalize the denominator and simplify.

5 / 5 + √10x
( 5 divided by 5 plus the square root of 10x)

Answer 1

is it
\[\large \frac{5}{5+\sqrt{10x}}\]

Answer 2

Yes!

Answer 3

ok then multiply top and bottom by
\[5-\sqrt{10x}\] and this works because
\[(a+b)(a-b)=a^2-b^2\] so \[5+\sqrt{10x})(5-\sqrt{10x})=25-10x\]

Answer 4

the denominator will be \(25-10x\) and the numerator will be
\[5(5+\sqrt{10x})\]

Answer 5

Yes I have done that already and my final answer was 5 - √10x / 5 - √10x.

I got this by trying to simplify so I divided 5 by 5 and got just 5 - √10x then i did the same to the bottom but it was 25 divided by 5 and i got 5 - √10x.

Answer 6

i.e. you will get
\[\frac{5(5+\sqrt{10x})}{25-10x}\] now you can factor a \(5\) out of the bottom and cancel

Answer 7

ok i think i have confused you
lets go slow

Answer 8

Okay sounds good! :)

Answer 9

What first?

Answer 10

\[\large \frac{5}{5+\sqrt{10x}}\] is the expression

Answer 11

the conjugate of \(5+\sqrt{10x}\) is \(5-\sqrt{10x}\) so we write
\[\large \frac{5}{5+\sqrt{10x}}\times \frac{5-\sqrt{10x}}{5-\sqrt{10x}}\]

Answer 12

leave the numerator in factored form i.e. do NOT multiply out out

Answer 13

you get
\[\large \frac{5(5-\sqrt{10x})}{25-10x}\]
is it clear how i got the denominator?

Answer 14

Yes very clear! I can do it that far just get confused when I try and simplify.

Answer 15

ok that was the hard part, now it the easy part
factor the denominator
\[\large \frac{5(5-\sqrt{10x})}{5(5-2x)}\]

Answer 16

How did you get 2x?

Answer 17

\[\large \frac{\cancel5(5-\sqrt{10x})}{\cancel5(5-2x)}=\frac{5-\sqrt{10x}}{5-2x}\]

Answer 18

oh how did i get the \(2x\) ?

Answer 19

the denominator was \(25-10x\) right?

Answer 20

Yes

Answer 21

each term has a common factor of \(5\) right?

Answer 22

Yes so did you just factor out 5?

Answer 23

yup

Answer 24

\[25-10x=5(5-2x)\] is all

Answer 25

which as math teachers would say "cancels" with the \(5\) in the numerator

Answer 26

Okay! Yay! That was what I was missing the entire time. So in any problem similar to this one if they can both be factored by the same number do I do that? Or was it just this one?

Answer 27

\[\large \frac{5(5-\sqrt{10x})}{25-10x}=\large \frac{5(5-\sqrt{10x})}{5(5-2x)}=\large \frac{\cancel5(5-\sqrt{10x})}{\cancel5(5-2x)}=\frac{5-\sqrt{10x}}{5-2x}\]

Answer 28

this was cooked up to do that
in general nothing cancels

Answer 29

now i have a question

Answer 30

Okay shoot.

Answer 31

do you really go to an Aviation Academy?

Answer 32

Yes I do. Ha. Are you asking because a person going to an aviation school should know how to do this?

Answer 33

no because you said in your profile you were
i don't want the pilot of my 787 worrying about square roots ever
just concentrate on getting there safely!

Answer 34

ooooh i see it is a high school
where attitude meets altitude lol

Answer 35

Haha very true! Although I don't want to be a pilot. I am going into the medical field and one of my dreams is to fly for AeroMed or something similar. Which is still aviation but I have not decided if I want to fly the helicopter or be the RN in it.

Thank you so much for helping me out! I really appreciate it.

Answer 36

yw