Question

lesson: simplifying radicals
simplify: √8
= √□ · √□
= □√□

Explanation:

Step1: Factor 8 into perfect square and other

We know that \( 8 = 4\times2 \), where 4 is a perfect square. So, \( \sqrt{8}=\sqrt{4\times2} \). By the property of square roots \( \sqrt{ab}=\sqrt{a}\cdot\sqrt{b} \) (for \( a\geq0,b\geq0 \)), we can write \( \sqrt{4\times2}=\sqrt{4}\cdot\sqrt{2} \).

Step2: Simplify \( \sqrt{4} \)

Since \( 2^2 = 4 \), \( \sqrt{4}=2 \). So, \( \sqrt{4}\cdot\sqrt{2}=2\sqrt{2} \).

Answer:

First two boxes: 4, 2 (in any order for the first step's square roots); then the next two boxes: 2, 2 (the coefficient and the remaining radical part). So filling in:
\( \sqrt{8}=\sqrt{\boldsymbol{4}}\cdot\sqrt{\boldsymbol{2}} = \boldsymbol{2}\sqrt{\boldsymbol{2}} \)