Simplify Radical Expression - Help

[mcruz65]

This is the way I approached this problem. Is it correct?
?(2-4/z^2 +2/z^4 )

?(1 4/z^4 -8/z^4 +2/z^4 )

?((-2)/z^4 )=z^2 ?(-2)

 

[tutor_joel]

nope, find the common denominator of z^4 and then you can add the fractions together. The denominator will come out. Looks like you forgot to multiply the numerators by the z variables when you found the common denominator. LaTex coming up...

 

[tutor_joel]

\(\displaystyle \sqrt{2-\frac{4}{z^2}+\frac{2}{z^4}}=\sqrt{\frac{2z^4-4z^2+2}{z^4}}=\frac{1}{z^2}\sqrt{2(z^4-2z^2+1)\)

Make sense? You got it from here?

 

[mcruz65]

OK, I see where I went wrong.

1/(z^2)sqrt(2(z^4-2z^2+1) What am I suppose to do with this?

Do I do the following:
sqrt(2), sqrt(z^4), sqrt(-2z^2), sqrt(1)

sqrt(2), 2, z sqrt(-2), sqrt(1)

 

[Denis]

Sorry Mr Cruz, but you have no idea what you're doing...
you need classroom help, which can't be provided here;
are you learning on your own, or attending math classes ?

 

[mcruz65]

Thanks for your help. I am a freshman in college and it has been a while since I been in school. I beg to differ, about how much I know. I have a full load of classes plus a family to support. Again thanks for your help.

 

[mmm4444bot]

?

1/(z^2) sqrt(2(z^4 - 2z^2 + 1)


sqrt(2), sqrt(z^4), sqrt(-2z^2), sqrt(1)

We can't break the radical up into all of those smaller radicals because the terms are being added, not multiplied.

If the terms were actually factors, instead (i.e., multiplied together), then there is a property of radicals for that situation.

\(\displaystyle \sqrt{(2)(z^4)(-2z^2)(1)} \ = \ \sqrt{2} \cdot \sqrt{z^4} \cdot \sqrt{-2z^2} \cdot \sqrt{1}\)

In your situation, we can't use that property on the polynomial. We can only separate the two factors of the radicand.

\(\displaystyle \sqrt{(2)(z^4 - 2z^2 + 1)} \ = \ \sqrt{2} \cdot \sqrt{z^4 - 2z^2 + 1}\)

Now, if the square root with the polynomial for a radicand is to simplify, then that polynomial must have at least one square factor, yes? I mean, we can only simplify squares underneath a square root sign.

So, the next step is to factor the polynomial z^4 - 2z^2 + 1, to see whether or not it contains any factors that are squares.

 

[mcruz65]

sqrt(z^4-2z^2+1)

sqrt(z^2-1)^2

z^2-1

sqrt(2) z^2 - 1

Is the final answer is 1/z^2 sqrt(2) z^2-1?

 

[tutor_joel]

\(\displaystyle \sqrt{2-\frac{4}{z^2}+\frac{2}{z^4}}=\sqrt{\frac{2z^4-4z^2+2}{z^4}}=\frac{1}{z^2}\sqrt{2(z^4-2z^2+1)}=\frac{1}{z^2}\sqrt{2(z^2-1)^2}=\frac{z^2-1}{z^2}\sqrt{2}=\sqrt{2}-\frac{\sqrt{2}}{z^2}\)

Up to you how you want to write the final answer. Either of the last two are probably fine