Question

which expression is equivalent to \\(\sqrt{336}\\) written in simplest radical form?\
\\(\bigcirc\\ 2\sqrt{84}\\)\
\\(\bigcirc\\ 84\sqrt{2}\\)\
\\(\bigcirc\\ 4\sqrt{21}\\)\
\\(\bigcirc\\ 21\sqrt{4}\\)

Explanation:

Step1: Factor 336

Find the largest perfect square factor of 336. We know that \( 16\times21 = 336 \), and 16 is a perfect square (\( 4^2 = 16 \)).
So, \( \sqrt{336}=\sqrt{16\times21} \).

Step2: Use radical property

Using the property \( \sqrt{ab}=\sqrt{a}\times\sqrt{b} \) (for \( a\geq0, b\geq0 \)), we get:
\( \sqrt{16\times21}=\sqrt{16}\times\sqrt{21} \).

Step3: Simplify \( \sqrt{16} \)

Since \( \sqrt{16} = 4 \), then \( \sqrt{16}\times\sqrt{21}=4\sqrt{21} \).

Now let's check the other options:

• For \( 2\sqrt{84} \), \( 84 = 4\times21 \), so \( 2\sqrt{84}=2\sqrt{4\times21}=2\times2\sqrt{21}=4\sqrt{21} \), but this is not in simplest form as 84 still has a perfect square factor.
• For \( 84\sqrt{2} \), \( 84\sqrt{2}=\sqrt{84^2\times2}=\sqrt{7056\times2}=\sqrt{14112}

eq\sqrt{336} \).

• For \( 21\sqrt{4} \), \( 21\sqrt{4}=21\times2 = 42\), and \( \sqrt{336}\approx18.33

eq42 \).

Answer:

C. \( 4\sqrt{21} \) (assuming the third option is labeled C, as per the order: first option \( 2\sqrt{84} \) (A), second \( 84\sqrt{2} \) (B), third \( 4\sqrt{21} \) (C), fourth \( 21\sqrt{4} \) (D))