Why does <strike>$(2^2\cdot 3^3)^2 = (2^2\cdot 3^3\cdot 4^4$)</strike>? nevermind, my bad

Question

I was messing around and I noticed that $(2^2\cdot3^3)^2 = 2^2\cdot 3^3\cdot 4^4$.

May sound strange but $108$ is a mystical number of ancient India, and was trying to deduce why, when I noticed it is $2^2\cdot 3^3$. So then I wondered what would happen if I multiplied that to $4$ to the $4^{\textrm{th}}$ power and discovered that it is $108^2$.

This got me thinking that maybe this is a universal formula that I just was not aware of... does $(a^2\cdot b^3)^2 = a^2\cdot b^3\cdot c^4$?

Pardon my ignorance, it has been many years since I've been in a math class.

Thanks.

UPDATE : my math was wrong, please disregard this question .. should of double-checked the calculation before posting

Answer 1

$2^2*3^3*4^4=4*27*256=27648\\(2^2*3^3)^2=108^2=11664$

Answer 2

Using the basic laws of exponents, one has

$$(a^2\cdot b^3)^2=(a^2\cdot b^3) \cdot(a^2\cdot b^3) = a^2 \cdot b^3\cdot (a^2\cdot b^3)$$

and, generally speaking, the expression $(a^2\cdot b^3)$ is not typically a fourth power of another integer $c$. Of course, it is the fourth power of $\sqrt[4]{a^2\cdot b^3}$, but that's not usually an integer.

So -- no.

Answer 3

$$(2^2\times 3^3)^2 = 2^23^34^4$$ $$\implies2^23^3 = 4^4 \text{, but }$$ this is impossible since LHS is odd, RHS is even