I noticed that it's possible to factor out the greatest factor from an expression raised to power without having to resolve the power first.
For example,
$(3x^4 +15x^2)^2 = (3x^2)^2\cdot(x^2+5)^2 = 9x^8+90x^6+225x^4$
Which is equivalent to having resolved the power first
$(3x^4 + 15x^2)^2 =(3x^4 + 15x^2) \cdot (3x^4 + 15x^2) = 9x^8+90x^6+225x^4$
In general, it seems that
$(ca + cb)^n = c^n (a+b)^n$
Where $a, b, c$ are monomials and $n$ is a constant
Also, it would seem that if we have two factors both raised to the same power then we can distribute.
$c^n (a+b)^n = (ca + cb)^n$
Where $a, b, c$ are monomials and $n$ is a constant
I'm wondering why this appears to be true as in what properties does this observation follow from?
Best Answer
This is rule 2 as explained here:
http://www.mclph.umn.edu/mathrefresh/exponents2.html
This is just rearranging the terms being multiplied in a product. For example:
$$(ab)^3 = (ab)*(ab)*(ab)$$
I can rearrange the a's and b's to bring the a's together and b's together. This follows from the associative and commutative properties of multiplication.
$$(ab)*(ab)*(ab) = (a*a*a)*(b*b*b) = a^3b^3$$