Squares and square roots

When we square #\sqrt{4}#, we get:

\[\left(\sqrt{\blue4}\right)^2=2^2=\blue4\]

In general, we can state:

The square of a square root is equal to the number inside the radical symbol.

In the same way, it also holds that:

#\begin{array}{rclrcl}\sqrt{\blue4^2}&=&\sqrt{16}&=&\blue4\\\sqrt{(\blue{-4})^2}&=&\sqrt{16}&=&\blue4\end{array}#

In general, we can state:

The square root of a square is equal to the absolute value of the number that is squared.

Examples

\[\begin{array}{rclrcl}\left(\sqrt{\blue2}\right)^2&=&\blue2 \\ \\ \left(\sqrt{\blue{20}}\right)^2&=&\blue{20} \\ \\ \\ \sqrt{\blue2^2}&=&\abs{\blue2}&=&\blue2 \\ \\ \sqrt{\blue{20}^2}&=&\abs{\blue{20}}&=&\blue{20} \\ \\ \sqrt{(\blue{-10})^2}&=&\abs{\blue{-10}}&=&\blue{10}\end{array}\]

Products of square roots

When we multiply #\sqrt{\blue4}# and #\sqrt{\green9}#, we get:

\[\sqrt{\blue4} \times \sqrt{\green9}=2 \times 3=6=\sqrt{36}=\sqrt{\blue4 \times \green9}\]

In general, we can state:

The product of two square roots is equal to the square root of the product of the numbers inside the radical symbols.

We can use this rule the other way around as well:

The square root of a product is the product of the square roots.

Examples

\[\begin{array}{rcl}\sqrt{\blue2}\times \sqrt{\green8}&=&\sqrt{\blue2 \times \green8} \\ &=& \sqrt{16} \\ &=& 4 \\ \\ \sqrt{12}&=&\sqrt{\blue4 \times \green3} \\&=& \sqrt{\blue4} \times \sqrt{\green3}\\&=&2\sqrt{3} \end{array}\]