# What is SQUARE ROOT OF 3 times the SQUARE ROOT of 21?

Sep 6, 2015

$\sqrt{3} \cdot \sqrt{21} = 3 \sqrt{7}$

#### Explanation:

You know that for real, positive numbers you have

color(blue)(sqrt(a) * sqrt(b) = sqrt(a * b)

This means that you can write

$\sqrt{3} \cdot \sqrt{21} = \sqrt{3 \cdot 21}$

If you write $21$ as the product of its prime factors $3$ and $7$, you can say that

$\sqrt{3 \cdot 21} = \sqrt{3 \cdot 3 \cdot 7} = \sqrt{9 \cdot 7}$

Now use the same multiplication property of radicals to write

sqrt(9 * 7) = sqrt(9) * sqrt(7) = 3 * sqrt(7) = color(green)(3sqrt(7)