Question

lesson: simplifying radicals
simplify: √147
= √□ · √□
= □√□

Explanation:

Step1: Factor 147 into perfect square and other

We know that \( 147 = 49\times3 \), where \( 49 \) is a perfect square (\( 7^2 \)). So we can write \( \sqrt{147}=\sqrt{49\times3} \). By the property of square roots \( \sqrt{ab}=\sqrt{a}\cdot\sqrt{b} \) (for \( a\geq0,b\geq0 \)), we get \( \sqrt{147}=\sqrt{49}\cdot\sqrt{3} \).

Step2: Simplify \( \sqrt{49} \)

Since \( 7^2 = 49 \), \( \sqrt{49}=7 \). So substituting back, we have \( 7\sqrt{3} \).

Answer:

First two blanks: \( 49 \), \( 3 \); last two blanks: \( 7 \), \( 3 \) (i.e., \( \sqrt{147}=\sqrt{49}\cdot\sqrt{3}=7\sqrt{3} \))