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

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The discussion focuses on simplifying the expression (ax + b)^(-1/2) - [sqrt{ax + b}]/b using rational exponents. The initial approach involves rewriting the expression with rational exponents and factoring out terms with smaller exponents. A participant mentions using b^0 as a placeholder to facilitate factoring, highlighting that this technique is a matter of personal preference. The conversation emphasizes that different methods of simplification can vary based on individual style. Overall, the thread illustrates the complexities of factoring and simplifying algebraic expressions.
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(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..
 

Thread 'Area of Overlapping Squares'

Here is a little puzzle from the book 100 Geometric Games by Pierre Berloquin. The side of a small square is one meter long and the side of a larger square one and a half meters long. One vertex of the large square is at the center of the small square. The side of the large square cuts two sides of the small square into one- third parts and two-thirds parts. What is the area where the squares overlap?
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