How to factor out s and distribute the power of 6

[umzung]

Homework Statement

How does one bracket become the other?

Relevant Equations

See the attempt at a solution.

How does

become

?

I can see the s has been factored out and the power of 6 distributed, but how do we know this happens, short of multiplying out the brackets?

 

[WWGD]

First factor the s in the top expression. Then what happens with the power 6? I think it will be clearer then.

 

[Mark44]

Homework Statement: How does one bracket become the other?
Homework Equations: See the attempt at a solution.

How does View attachment 253352become View attachment 253353?

I can see the s has been factored out and the power of 6 distributed, but how do we know this happens, short of multiplying out the brackets?

Just factor the expression inside the parentheses.
$\frac 1 2 s + \frac 1 2 s^2 = s( \frac 1 2 + \frac 1 2 s)$.

Then use the property that $(ab)^n = a^nb^n$

 

[umzung]

Just factor the expression inside the parentheses.
$\frac 1 2 s + \frac 1 2 s^2 = s( \frac 1 2 + \frac 1 2 s)$.

Then use the property that $(ab)^n = a^nb^n$

That's clearer, thanks.

 

[HallsofIvy]

I would actually go a little further than that. $$\frac{1}{2}s+ \frac{1}{2}s^2= \frac{1}{2}s(1+ s)$$ so that $$\left(\frac{1}{2}s+ \frac{1}{2}s^2\right)^6= \frac{1}{64}s^6(1+ s)^6$$.