How do you apply one prime factorisation to another?

Question

I have a question in my paper, Express 4225 as the product of its prime factors in index notation. That was easy to answer, but the next question is express the square root of 42250000 using prime factorisation. Apparently there is a way to use my answer in the first question to do the second, but how do I?

Answer 1

By way of $$ 42\,250\,000 = 4225\cdot 10\,000 $$ the two numbers have a lot of factorisation in common.

Answer 2

Using that $\sqrt{x}=x^{\frac{1}{2}}$, index laws and the prime factorisations of $1000$ and $4255$, as well as that $42250000=4225\times10000$, we have;

$\sqrt{42250000}=\sqrt{4225}\sqrt{10000}=\sqrt{5^{2}13^{2}}\sqrt{10^4}=(5^{2}13^{2})^{\frac{1}{2}}({10^4})^{\frac{1}{2}}=5\times 13\times10^2$

This works because every factor in the prime factorisation of $42250000$ appears raised to a power which is a multiple of $2$, so the square root is guaranteed to be a whole number.

Answer 3

If $$n=p_1^{2k_1}\cdots p_m^{2k_m}$$ where the $k_i$ are non-negative integers then: $$\sqrt n=p_1^{k_1}\cdots p_m^{k_m}$$

Answer 4

Hint $\,f(ab) = f(a)f(b)\,$ where $\,f(n) := $ prime factorization of $n$ and, furthermore, if $\,a,b\,$ are coprime then $\,ab\,$ is a square $\iff a,b$ are squares $\iff$ every prime occurs to even power in their prime factorizations, all being true by FTA = existence and uniqueness of prime factorizations (all inferences fail lacking such).