Question

multiply. assume u, v, and w are greater than or equal to zero, and write your answer in simplest form.\\(\sqrt{3u^{2}v^{4}}\cdot\sqrt{30w^{4}}\\)

Explanation:

Step1: Use property of square roots ($\sqrt{a}\cdot\sqrt{b}=\sqrt{ab}$)

$\sqrt{3u^{2}v^{4}}\cdot\sqrt{30w^{4}}=\sqrt{(3u^{2}v^{4})\cdot(30w^{4})}$

Step2: Multiply the terms inside the square root

First, multiply the coefficients: $3\times30 = 90$. Then, the variables: $u^{2}v^{4}w^{4}$. So we get $\sqrt{90u^{2}v^{4}w^{4}}$

Step3: Simplify the square root

Factor 90: $90 = 9\times10$. So $\sqrt{9\times10\times u^{2}\times v^{4}\times w^{4}}$. Now, take square roots of perfect squares: $\sqrt{9}=3$, $\sqrt{u^{2}} = u$ (since $u\geq0$), $\sqrt{v^{4}}=v^{2}$, $\sqrt{w^{4}}=w^{2}$. So we have $3u v^{2}w^{2}\sqrt{10}$

Answer:

$3uv^{2}w^{2}\sqrt{10}$