Algebra: Calculating with exponents and roots
Calculating with square roots
The following rules are very convenient when simplifying expressions with square roots.
\[\sqrt{\blue a \cdot \green b}=\sqrt{\blue a} \cdot \sqrt{\green b}\] |
Example \[\begin{array}{rcl}\sqrt{\blue 4 \cdot \green x}&=&\sqrt{\blue 4} \cdot \sqrt{\green x}\\ &=& 2 \cdot \sqrt{\green x} \end{array}\] |
\[\sqrt{\frac{\blue a}{\green b}}=\frac{\sqrt{\blue a}} {\sqrt{\green b}}\] |
Example \[\begin{array}{rcl}\sqrt{\dfrac{\blue 4}{\green x}}&=&\dfrac{\sqrt{\blue 4}}{\sqrt{\green x}}\\ &=& \dfrac{2}{\sqrt{\green x}} \end{array}\] |
For #\blue a \geq 0# we have: \[\sqrt{\blue a^2}=\blue a\] and \[\left(\sqrt{\blue a}\right)^2=\blue a\] |
Examples \[\begin{array}{rcl}\sqrt{\blue x^4}&=&\sqrt{\blue x^2 \cdot \blue x^2}\\ &=& \sqrt{\blue x^2} \cdot \sqrt{\blue x^2} \\ &=& \blue x \cdot \blue x \\&=&\blue x^2\\ \\ \left(\sqrt{\blue x}\right)^4 &=& \left(\left(\sqrt{\blue x}\right)^2\right)^2 \\ &=& \blue x^2\end{array}\] |
#\begin{array}{rcl}
\sqrt{64 \cdot c^{2} \cdot y^{2}}&=& \sqrt{64} \cdot \sqrt{c^{2}} \cdot \sqrt{y^{2}} \\
&& \phantom{xxx}\blue{\text{rule }\sqrt{a \cdot b} = \sqrt{a} \cdot \sqrt{b}}\\
&=& 8 \cdot c \cdot y\\
&& \phantom{xxx}\blue{\text{rule }\sqrt{a^2}=a \text{ and square root calculated }}
\end{array}#
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