Question

multiply. assume n is greater than or equal to zero, and write your answer in simplest form. $sqrt{7n^{3}}cdotsqrt{7}$

Explanation:

Step1: Use product - rule of radicals

The product - rule of radicals states that $\sqrt{a}\cdot\sqrt{b}=\sqrt{ab}$ for $a\geq0$ and $b\geq0$. Here, we have $\sqrt{7n^{3}}\cdot\sqrt{7}$. So, $\sqrt{7n^{3}}\cdot\sqrt{7}=\sqrt{7n^{3}\times7}$.

Step2: Simplify the expression inside the square - root

Multiply the numbers: $7n^{3}\times7 = 49n^{3}$. So, we get $\sqrt{49n^{3}}$.

Step3: Break down the square - root

We know that $\sqrt{49n^{3}}=\sqrt{49}\cdot\sqrt{n^{3}}$. Since $\sqrt{49}=7$ and $\sqrt{n^{3}}=\sqrt{n^{2}\cdot n}=n\sqrt{n}$ (using the rule $\sqrt{ab}=\sqrt{a}\cdot\sqrt{b}$ with $a = n^{2}$ and $b = n$ and $\sqrt{n^{2}}=n$ for $n\geq0$).

Step4: Combine the results

$7\cdot n\sqrt{n}=7n\sqrt{n}$.

Answer:

$7n\sqrt{n}$