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

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Discussion Overview

The discussion revolves around finding the factorial expression for the mathematical expression (ax + b)-1/2 - [sqrt{ax + b}]/b. The focus is on algebraic manipulation and factoring techniques, with participants exploring different approaches to simplify the expression.

Discussion Character

  • Exploratory, Technical explanation, Mathematical reasoning

Main Points Raised

  • One participant suggests starting with rational exponents to rewrite the expression, indicating a method for simplification.
  • Another participant questions the introduction of b0 in the expression, seeking clarification on its purpose.
  • A further reply explains that b0 serves as a placeholder for factoring, emphasizing that such choices can be a matter of personal preference in simplification.
  • One participant indicates they will return to the problem later, suggesting ongoing engagement with the topic.

Areas of Agreement / Disagreement

Participants do not reach a consensus on the necessity of certain steps in the factoring process, and there is no resolution on the best approach to take.

Contextual Notes

The discussion includes personal preferences in factoring methods, which may influence how participants approach the problem. There are also unresolved questions about the role of certain terms in the expressions.

mathdad
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Factor

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

This one is tricky. Can someone get me started?
 
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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
 
Where did b^0 come from?
 
RTCNTC said:
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
 
I will work on this later..
 

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