Numbers: Powers and roots
Rules of calculation for roots
For roots, there are some important rules of calculation. We will use these later to write the root as simply as possible.
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
\[\begin{array}{rcl}\sqrt{\blue2}\times \sqrt{\green8}&=&\sqrt{\blue2 \times \green8} \\ &=& \sqrt{16} \\ &=& 4 \\ \\ \sqrt{12}&=&\sqrt{\blue4 \times \green3} \\&=& \sqrt{\blue4} \times \sqrt{\green3}\\&=&2\sqrt{3} \end{array}\]
The calculation rule states that the square of a root equals the number inside the radical symbol. This means that:
\[\left(\sqrt{71}\right)^2=71\]
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