Algebra: quick explanation needed :)

[uffechristian]

Please tell me how they went from first step to second step.
thanks in advance.

 

[Deleted member 4993]

Please tell me how they went from first step to second step.
thanks in advance.View attachment 9969

I'll do a similar problem. Do we see that we can write:

x * y + y³ ............Factoring out 'y' we get

= y * (x + y²)

 

[HallsofIvy]

Do you mean from \(\displaystyle \frac{x(1- 2x)^{-3/2}+ (1- 2x)^{-1/2}}{1- x}\) to \(\displaystyle (1- 2x)^{-3/2}\frac{x+ [x+ (1- 2x)^1]}{1- x}\)?

It says "factor out the power of 1- 2x to the smallest exponent". The two exponents of 1- 2x are -3/2 and -1/2. Of those two -3/2 is the smaller.

Factoring \(\displaystyle (1- 2x)^{-3/2}\) out of \(\displaystyle x(1- 2x)^{-3/2}\) leaves, of course, \(\displaystyle \frac{x(1- 2x)^{-3/2}}{(1- 2x)^{-3/2}}= x\) and factoring it out of \(\displaystyle (1- 2x)^{-1/2}= \frac{1- 2x)^{-1/2}}{(1- 2x)^{-3/2}}= (1- 2x)^{-1/2+ 3/2}= 1- 2x\).

So \(\displaystyle x(1- 2x)^{-3/2}+ (1- 2x)^{-1/2}= (1- 2x)^{-3/2}(x+ (1- 2x))\). Of course the denominator, 1- x, doesn't change.

 

[mmm4444bot]

Please tell me how they went from first step to second step.

You didn't explain why you're confused, so we have to guess. It seems like you're asking how they did the first step, not the second. (The second step simply removes grouping symbols around 1-2x.)

The first step changes the left-hand side into the right-hand side (on the first line shown). To complete this step, they followed the instruction printed in the right margin; they factored out the power of (1-2x) having the smallest exponent. On the left-hand side, we see the exponents are -3/2 and -1/2. The smaller number is -3/2, so they factored out (1-2x)^(-3/2).

Let's use u-substitution, to simplify the notation, and let's ignore the denominator (it's not changing). Perhaps, this will help you see the first step.

u = 1 - 2x

Rewriting the first step, we have:

x∙u^-3/2 + u^-1/2 = u^-3/2 ∙ [x + u¹]

The two exponents shown in red must sum to -1/2 because we add exponents when multiplying powers having the same base, like this:

u^-3/2 ∙ u^2/2 = u^-1/2

Because: -3/2 + 2/2 = -1/2

If you're still unsure, please explain why and ask specific questions. Cheers :cool: