Simplify an Equation

[Hockeyman]

How would you simplify this equation?

4(k/SqRt of 2 times k/SqRt of 2) + 4(k/SqRt of 2 times k) + k^2
--------------------------
2
Divide just what is in the parentheses by 2.


I hope this isn't too confusing, I didn't know how to do the square root sign. I know the answer is (2+2 times the SqRt of 2) k^2

Thanks for any help in advance.

 

[soroban]

Hello, Hockeyman!

It took me several tries to get your drift . . .

\(\displaystyle \L\frac{4\left(\frac{k}{\sqrt{2}}\cdot\frac{k}{\sqrt{2}}\right)\,+\,4\left(\frac{k}{\sqrt{2}}\cdot k\right)}{2}\,+\,k^2\)

\(\displaystyle \L\;\;\;=\;\frac{\frac{4k^2}{2}\,+\,\frac{4k^2}{\sqrt{2}}}{2}\,+\,k^2\)

\(\displaystyle \L\;\;=\;\frac{2k^2\,+\,2\sqrt{2}k^2}{2}\,+\,k^2\)

\(\displaystyle \L\;\;=\;k^2\,+\,\sqrt{2}k^2\,+\,k^2\)

\(\displaystyle \L\;\;=\;2k^2\,+\,\sqrt{2}k^2\)

\(\displaystyle \L\;\;=\;(2\,+\,\sqrt{2})k^2\)

 

[Hockeyman]

I'm sorry I should have been more specific, the divided 2 should only be under the first set of parentheses not the whole thing. It didn't come out on the screen quite how i typed it. And the answer should be (2+2 times the square root of 2) k^2. In your answer there is a +2 missing.

 

[Unco]

\(\displaystyle \Huge \mbox{4\cdot\frac{\left(\frac{k}{\sqrt{2}} \times \frac{k}{\sqrt{2}}\right)}{2} + 4\left(\frac{k}{\sqrt{2}} \times k\right) + k^2\)

\(\displaystyle \Huge \mbox{ = \frac{4}{2}\left(\frac{k^2}{\sqrt{2}\sqrt{2}}\right) + 4\left(\frac{k^2}{\sqrt{2}}\right) + k^2}\)

\(\displaystyle \Huge \mbox{ = 2\left(\frac{k^2}{2}\right) + \frac{4k^2}{\sqrt{2}} + k^2}\)

\(\displaystyle \Huge \mbox{ = k^2 + \frac{4k^2}{\sqrt{2}} + k^2}\)

\(\displaystyle \Huge \mbox{ = k^2 + \frac{4\cdot\sqrt{2}k^2}{2} + k^2}\)

\(\displaystyle \Huge \mbox{ = 2k^2 + 2\sqrt{2}k^2}\)

\(\displaystyle \Huge \mbox{ = \left(2 + 2\sqrt{2}\right)k^2}\)

 

[Hockeyman]

Thank you for your detailed explanation. I do have a few questions however.

On step one how did you get 4/2?

On step four how did you get from 4k^2/SqRt of 2 --> 4 times the SqRT of 2k^2/2

And how did you get from you last step to your answer?

Sorry to be so much trouble just had some difficulty following your steps.

 

[Unco]

On step one how did you get 4/2?

Perhaps this is what you mean:

\(\displaystyle \L \mbox{ 4\cdot\frac{C}{2} = 4\cdot\frac{1}{2}C = \frac{4}{2}C}\)

On step four how did you get from 4k^2/SqRt of 2 --> 4 times the SqRT of 2k^2/2

We rationalise the denominator. That is, manipulating the fraction such that the denominator is rational (doesn't have square roots and so forth).

\(\displaystyle \L \mbox{ sqrt{2} \cdot \sqrt{2} = 2^{\frac{1}{2}} \cdot 2^{\frac{1}{2}} = 2^{\frac{1}{2} + \frac{1}{2}} = 2^1 = 2\), so if we multiply denominator by \(\displaystyle \mbox{\sqrt{2}}\), we will make it rational (as 2 is rational).

What we multiply the denomiantor by we must mutliply the numerator by as well (so we are effectively multiplying the whole fraction by 1*).

\(\displaystyle \L \mbox{ \frac{4k^2}{\sqrt{2}} = \frac{4k^2}{\sqrt{2}} \cdot \frac{\sqrt{2}}{\sqrt{2}} = \frac{4\sqrt{2}k^2}{\sqrt{2}\sqrt{2}} = \frac{4\sqrt{2}k^2}{2}}\)

* \(\displaystyle \L \frac{ \sqrt{2}}{\sqrt{2}} = 1\)

And how did you get from you last step to your answer?

\(\displaystyle \mbox{ 2k^2 + 2\sqrt{2}k^2}\)

There is a common factor of \(\displaystyle \mbox{2k^2}\), but I went with just \(\displaystyle \mbox{k^2\) because that was how the provided answer was written.

This is no different to
\(\displaystyle \mbox{ ab + ac = a(b + c)}\)