Need help figuring out how problem is simplified

[andy1212]

I have a math answer example which shows this, (and underlined numbers and/or variables means over or devided by)

step 1 = sqrt(x+h)-sqrt(x)
h

2 = sqrt(x+h)-sqrt(x) * sqrt(x+h)+sqrt(x)
h * sqrt(x+h)+sqrt(x)

3 = (sqrt(x+h)-sqrt(x)(sqrt(x+h)+sqrt(x))
h(sqrt(x+h)+sqrt(x))

4 = ((sqrt(x+h))^2-(sqrt(x))^2)
h(sqrt(x+h)+sqrt(x))

5 = (x+h-x)
h(sqrt(x+h)+sqrt(x))

6 = h
h(sqrt(x+h)+sqrt(x))

7 = 1
sqrt(x+h)+sqrt(x)

8 = lim h ->0 1
sqrt(x+h)+sqrt(x)

9 = 1
²sqrt(x)

and I'm wondering how they decide to take the whole numerator in step 1 and multiply it to the denominator and then also in step 2 why do they switch the -sqrt(x) to +sqrt(x)? Also in step 4 to 5 to 6 it arrives at h but how, they don't really show the steps they took, if I were to do it I guess would come up with this answer for the numerator of step 6,

I'm not sure how the square roots cancel out and what square root of x+h or x is but lets say I do know why they cancel out and then I show the next step, so I'm following BEDMAS and I'm left with ((x+h)^2 - (x)^2) = ((x+h)(x+h) - (x)(x)) = x^2+xh+xh+h^2 - x^2 = 2xh + h^2,

so I've already messed up there but I guess I'm just really confused with this problem and any help to explain it is appreciated! Thanks for your time and help.

 

[HallsofIvy]

That's a fairly standard method of "rationalizing the numerator/denominator". You may have learned in algebra or pre-calculus that a fraction like \(\displaystyle \frac{1}{\sqrt{2}}\) can be written like without the root in the denominator by multiplying both numerator and denominator by \(\displaystyle \frac{\sqrt{2}}{\sqrt{2}}\): \(\displaystyle \frac{1}{\sqrt{2}}\)\(\displaystyle \frac{\sqrt{2}}{\sqrt{2}}\)\(\displaystyle = \frac{(1)(\sqrt{2})}{(\sqrt{2})(\sqrt(2))}= \frac{\sqrt{2}}{2}\).

We often are more concerned with having rational terms (no radicals) in the denominator of a fraction since we need to get "common denominators" when adding fractions but here you have an example where we want no radicals in the numerator.

If you have something like \(\displaystyle \sqrt{a}- \sqrt{b}\) in the numerator/denominator, use the fact that \(\displaystyle (a- b)(a+ b)a= a^2- b^2\) so that \(\displaystyle (\sqrt{a}- \sqrt{b})(\sqrt{a}+ \sqrt{b}) = (\sqrt{a})^2- (\sqrt{b})^2= a- b.\) That was what was being used here. The numerator of your fraction was \(\displaystyle \sqrt{x+ h}- \sqrt{x}\) so multiplying by \(\displaystyle \sqrt{x+ h}+ \sqrt{x}\) converts that to \(\displaystyle (x+ h)- x= h\) just as you show.

 

[andy1212]

Thanks so much! I'm watching some videos on youtube explaining rationalizing and conjugating and it's starting to make sense now. I've been out of school for years and just upgrading my math mark online so this was a big help!

 

[andy1212]

Thanks Denis, that makes it even easier

 

[andy1212]

Ok I figured it out at the part where i was stuck as to how they arrive with the answer they did but still have a couple questions,

= (sqrt(x+h))^2-(sqrt(x))^2 = (sqrt(1x+1h))^2-(sqrt(1x))^2
square root of 1 is 1 and 1 mupltiply 1 is 1 which would equal
= x+h-x

I was just wondering if we broke it down into even further steps, after gettin the square roots inside the brackets then we would expand the squared parts like this (x+h)(x+h)-(x)(x). The part above shows us skipping this part and just squaring each variable individually, but shouldnt we have factored? just confused on that.