Question

lesson: simplifying radicals
simplify: \\(\sqrt{18}\\)
\\(= \sqrt{\square} \cdot \sqrt{\square}\\)
\\(= \square \sqrt{\square}\\)

Explanation:

Step1: Factor 18 into 9 and 2

We know that \( 18 = 9\times2 \), and since \( \sqrt{ab}=\sqrt{a}\cdot\sqrt{b} \) (for \( a\geq0,b\geq0 \)), we can write \( \sqrt{18}=\sqrt{9}\cdot\sqrt{2} \).

Step2: Simplify \( \sqrt{9} \)

We know that \( \sqrt{9} = 3 \) because \( 3\times3 = 9 \). So substituting back, we get \( \sqrt{18}=3\sqrt{2} \).

Answer:

First two boxes: 9, 2 (in either order for the first two square roots, but typically we use the perfect square factor first, so 9 and 2); last two boxes: 3, 2. So filling in:
\( \sqrt{18}=\sqrt{\boldsymbol{9}}\cdot\sqrt{\boldsymbol{2}} = \boldsymbol{3}\sqrt{\boldsymbol{2}} \)