Question

the first two steps for simplifying \\(\sqrt{3,150}\\) are shown.\
step 1: \\(\sqrt{3,150} = \sqrt{7 \cdot 5 \cdot 5 \cdot 3 \cdot 3 \cdot 2}\\)\
step 2: \\(\sqrt{7 \cdot 5 \cdot 5 \cdot 3 \cdot 3 \cdot 2} = \sqrt{7 \cdot 5^2 \cdot 3^2 \cdot 2}\\)\
complete the last step by simplifying the expression. what is \\(\sqrt{3,150}\\) simplified?\
\\(15\sqrt{14}\\)\
\\(14\sqrt{15}\\)\
\\(14\\)\
\\(15\\)

Explanation:

Step 1: Recall square root property

For a square root \(\sqrt{ab}\) where \(a = m^2\) (a perfect square), we can use the property \(\sqrt{m^2n}=m\sqrt{n}\) (for \(m\geq0\), \(n\geq0\)). From Step 2, we have \(\sqrt{7\cdot5^{2}\cdot3^{2}\cdot2}\).

Step 2: Apply the square root property

We can split the square root into the product of square roots of perfect squares and the remaining factor: \(\sqrt{5^{2}\cdot3^{2}\cdot7\cdot2}=\sqrt{5^{2}}\cdot\sqrt{3^{2}}\cdot\sqrt{7\cdot2}\).

Since \(\sqrt{5^{2}} = 5\) and \(\sqrt{3^{2}}=3\), we multiply these together: \(5\times3 = 15\). Then we have \(15\sqrt{7\cdot2}=15\sqrt{14}\).

Answer:

\(15\sqrt{14}\) (corresponding to the option with "15√14")