O

#### OrbitalPower

##### Guest

**1. The problem statement, all variables and given/known data**

[tex]t[\frac{1}{2} (4- t)^\frac{-1}{2}(-1)] + (4 - t)^\frac{1}{2}[/tex]

The way I'd do it is

[tex]\frac{-t}{2(4-t)^\frac{1}{2}}[/tex] + [tex]\frac{\sqrt{4-t}}{1}}[/tex]

Get it over a common denominator

[tex]\frac{-t+\sqrt{4-t}*2\sqrt{4-t}}{2(4-t)^\frac{1}{2}}[/tex]

simplify:

[tex]\frac{-t + 2(4-t)}{2(4-t)^\frac{1}{2}}[/tex]

=

[tex]\frac{8-3t}{2(4-t)^\frac{1}{2}}[/tex]

The way I see books do it:

[tex]t[\frac{1}{2} (4- t)^\frac{-1}{2}(-1)] + (4 - t)^\frac{1}{2}[/tex]

=

[tex]\frac{1}{2}(4-t)^\frac{-1}{2}[-t+ 2(4-t)][/tex]

=

[tex]\frac{8-3t}{2(4-t)^\frac{1}{2}}[/tex]

What exactly is he sending through here? How does he know what to pull out in the second step? Is he mostly just skipping steps or is there a maneuver I'm missing here? I think I see the relationship between my third step (common denominator) and his second step, but I wouldn't know how to get that quickly without doing the algebra.

Here's another example of what I'm talking about:

x^2 * [ (1/2) * (1-x^2)^(-1/2) * (-2x)] + (1 - x^2)^(1/2) * (2x)

**1**

=

-x^3(1-x^2)^(-1/2) + 2x(1 - x^2)^(1/2)

**2**

=

x(1 - x^2)^(-1/2) * [-x^2(1) + 2(1 - x^2)]

**3**

=

x * (2 - 3x^2)/ (sqrt(1 - x^2))

**4**

Again, same questions. This actually looks confusing to me. What's going on at steps 2 and 3?

How did he know how to separate them like that without writing over a common factor. I've seen it done like this in other places, and it looks to me as if steps are being skipped.