Question

question 4 of 5

select the correct answer.

which calculation correctly uses prime factorization to write \\(\sqrt{240}\\) in simplest form?

\\(\bigcirc\\) \\(\sqrt{240} = \sqrt{2 \cdot 2 \cdot 2 \cdot 3 \cdot 5} = 2\sqrt{30}\\)
\\(\bigcirc\\) \\(\sqrt{240} = \sqrt{2 \cdot 2 \cdot 2 \cdot 3 \cdot 10} = 2\sqrt{60}\\)
\\(\bigcirc\\) \\(\sqrt{240} = \sqrt{2 \cdot 2 \cdot 2 \cdot 2 \cdot 3} = 4\sqrt{3}\\)
\\(\bigcirc\\) \\(\sqrt{240} = \sqrt{2 \cdot 2 \cdot 2 \cdot 2 \cdot 3 \cdot 5} = 4\sqrt{15}\\)

Explanation:

⚡ Using what you learned: simplifying radical expressions

Step 1: Find the prime factorization of 240

$$ 240 = 2 \cdot 120 = 2 \cdot 2 \cdot 60 = 2 \cdot 2 \cdot 2 \cdot 30 = 2 \cdot 2 \cdot 2 \cdot 2 \cdot 15 = 2 \cdot 2 \cdot 2 \cdot 2 \cdot 3 \cdot 5 $$

Step 2: Group the prime factors into pairs

$$ \sqrt{240} = \sqrt{(2 \cdot 2) \cdot (2 \cdot 2) \cdot 3 \cdot 5} $$

Step 3: Simplify the radical

$$ \sqrt{240} = 2 \cdot 2 \cdot \sqrt{3 \cdot 5} = 4\sqrt{15} $$

Answer:

The correct calculation is:

$$ \sqrt{240} = \sqrt{2 \cdot 2 \cdot 2 \cdot 2 \cdot 3 \cdot 5} = 4\sqrt{15} $$

(the fourth option)