Question

simplifying nonperfect roots
which expression is equivalent to \\(sqrt{120x}\\)?
options:
\\(2sqrt{30}\\)
\\(2sqrt{30x}\\)
\\(60sqrt{2x}\\)
\\(4sqrt{30x}\\)

Explanation:

Step1: Factor the radicand

We start with the expression $\sqrt{120x}$. First, we factor 120 into its prime factors. We know that $120 = 4\times30$, where 4 is a perfect square. So we can rewrite the square root as $\sqrt{4\times30x}$.

Step2: Apply the square root property

Using the property of square roots $\sqrt{ab}=\sqrt{a}\times\sqrt{b}$ (where $a\geq0$ and $b\geq0$), we can split the square root: $\sqrt{4\times30x}=\sqrt{4}\times\sqrt{30x}$.

Step3: Simplify the perfect square root

Since $\sqrt{4} = 2$, we substitute that back in: $2\times\sqrt{30x}=2\sqrt{30x}$.

Answer:

The expression equivalent to $\sqrt{120x}$ is $2\sqrt{30x}$ (the left - most option among the given choices).