Factoring out a greatest common factor from an expression raised to a power.

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?

• Are you referring to the property that $(a\cdot b)^n = a^n\cdot b^n$. Oct 9, 2020 at 20:22
• I mean, $(ca + cb)^n = \Big(c(a + b)\Big)^n = c^n (a+b)^n$ by the very property I refer to in my first comment. Just tak $x=c, y= (a+b)$, so $(x\cdot y)^n = x^ny^n$. Oct 9, 2020 at 20:45
• @amWhy Yes, that helps a lot. Thank you. Oct 10, 2020 at 2:52

$$(ab)^3 = (ab)*(ab)*(ab)$$
$$(ab)*(ab)*(ab) = (a*a*a)*(b*b*b) = a^3b^3$$