Tips for mastering algebraic simplification?

[EJnr]

Good Day!

I have the equation below that I am looking to simplify to the given form. I can see roughly the logic in the simplification but cannot complete it myself right now.

Rather than just being given the answer I would appreciate if the good users of this board would also be able to advise on how I can master these simplification techniques rather than having to ask for answers - any online resources that detail the rules of simplification, ideally with worked examples, is what I had in mind. Thank you in advance for your assistance, please find the equation and the simplified form that I am trying to reach below:

Regular equation: (1000(2+Q)^0.5 - 500Q(2+Q)^-0.5)/(2+Q)

Simplified form I am trying to reach: 2000 + 500Q/(2+Q)^1.5

I hope that the notation is clear - if anyone could also advise on the programs others are using to enter the standard mathematical notation that would also be much appreciated.

Best Wishes and Many Thanks!

EJnr

 

[galactus]

I will show you how I worked this one. Then, maybe you will know how to do subsequent problems like this.

Think of how you add or subtract any fraction.

Start with:

\(\displaystyle \frac{1000\sqrt{2+Q}-\frac{500Q}{\sqrt{2+Q}}}{2+Q}\)

Divide by 2+Q in the denominator:

\(\displaystyle \frac{1000}{\sqrt{2+Q}}-\frac{500Q}{(Q+2)^{\frac{3}{2}}}\)

Get the denominators the same by multiplying top and bottom of left side by \(\displaystyle Q+2\):

\(\displaystyle \frac{(Q+2)1000-500}{(Q+2)^{\frac{3}{2}}}\)

\(\displaystyle \frac{2000+500Q}{(2+Q)^{\frac{3}{2}}}\)

You can use LaTeX to post topics the way I did. Click on 'quote' in the upper right corner of this post to see the code I typed to make it display this way.

 

[soroban]

Hello, EJnr!

\(\displaystyle \text{Given the expressionb: }\;\frac{1000(2+Q)^{\frac{1}{2}} - 500Q(2+Q)^{\text{-}\frac{1}{2}}}{2+Q}\)

\(\displaystyle \text{Simplified form I am trying to reach: }\;\frac{2000 + 500Q}{(2+Q)^{\frac{3}{2}}}\)

\(\displaystyle \text{Multiply by: }\;\frac{(2+Q)^{\frac{1}{2}}}{(2+Q)^{\frac{1}{2}}}\)

. . \(\displaystyle \frac{(2+Q)^{\frac{1}{2}}}{(2+Q)^{\frac{1}{2}}} \cdot \frac{1000(2+Q)^{\frac{1}{2}} - 500Q(2+Q)^{\text{-}\frac{1}{2}}}{2+Q} \;\;=\;\;\frac{1000(2+Q) - 500Q}{(2+Q)^{\frac{3}{2}}} \;\;=\;\;\frac{2000 + 1000Q - 500Q}{(2+Q)^{\frac{3}{2}}} \;\;=\;\;\frac{2000 + 500Q}{(2+Q)^{\frac{3}{2}}}\)

 

[Denis]

(1000(2+Q)^0.5 - 500Q(2+Q)^-0.5) / (2+Q)

Let x = 2 + Q : I usually do this, less chances for errors/typos

[1000SQRT(x) - 500Q / SQRT(x)] / x

Carry on...