Prime Factorization for Square Roots with unknowns

Question

I need to help my daughter with math, but I don't understand it myself. We need to solve for $x$ and $y$ in the following equation, using prime factorization:

$$\sqrt{1890x} = \sqrt{2100y}$$

Can anyone show me how to do it?

Thanks.

Answer 1

If you square both sides, you have

$$1890x = 2100y$$.

Get the prime factorization of those numbers to write

$$2\cdot 3^3\cdot 5\cdot 7 x = 2^23\cdot 5^2 7 y.$$

Cancel all the common factors:

$$9x = 10 y.$$

If $a$ is any integer then $x=10a$, $y=9a$ will solve the equation. The square roots force $a$ to be positive.