MHB What is the Factorial Expression for (ax + b)^(-1/2) - [sqrt{ax + b}]/b?

[mathdad]

Factor

(ax + b)^(-1/2) - [sqrt{ax + b}]/b

This one is tricky. Can someone get me started?

 

[MarkFL]

I would begin by using rational exponents in place of the radical notation:

$$(ax+b)^{-\frac{1}{2}}-\frac{\sqrt{ax+b}}{b}=b^0(ax+b)^{-\frac{1}{2}}-b^{-1}(ax+b)^{\frac{1}{2}}$$

Next, continue, using the technique I explained in your other recent threads...you now have two factors in each expression in the given difference, so begin by factoring out those with the smaller exponents. :D

 

[mathdad]

Where did b^0 come from?

 

[MarkFL]

Where did b^0 come from?

That's just a placeholder to make factoring a little easier...once you get more practice you won't need it. Also, it's part of how I choose to factor, and not absolutely necessary. For example, if I have the expression:

$$a+\frac{a}{b}$$

Then I would choose to factor as:

$$\frac{a}{b}(b+1)$$

rather than:

$$a\left(1+\frac{1}{b}\right)$$

Much of simplifying expressions can simply be personal preference. :D

 

[mathdad]

I will work on this later..