Numbers: Powers and roots
Standard notation of roots
We have seen some calculation rules for roots and roots of fractions. Using these calculation rules, we can simplify roots.
When a root occurs in an expression, the expression is in standard notation if it meets the following requirements.
- The number in front of the radical sign is as large as possible.
This means that we eliminate the squares from the number inside the radical sign. Keep in mind that we only want a single square root. Therefore, you should not 'simplify' #\sqrt{14}# to #\sqrt{2}\cdot\sqrt{7}#. We do the following.
\[\sqrt{12}=\sqrt{2^2 \cdot 3}=\sqrt{2^2} \cdot \sqrt{3}=2\sqrt{3}\] - There are no fractions inside the radical sign. We use that the root of a fraction equals the fraction of the roots. \[\sqrt{\frac{2}{3}}=\frac{\sqrt{2}}{\sqrt{3}}\]
- There are no roots in the denominator of the fraction.
We achieve this by multiplying the numerator and the denominator of the fraction by the root from the denominator.
\[\frac{1}{\sqrt{3}}=\frac{\sqrt{3}}{\sqrt{3} \cdot \sqrt{3}}=\frac{\sqrt{3}}{3}\] - The rational numbers in the expression are simplified fractions. \[\frac{3}{\sqrt{3}}=\frac{3\sqrt{3}}{\sqrt{3} \cdot \sqrt{3}}=\frac{3\sqrt{3}}{3}=\sqrt{3}\]
Examples
\[\begin{array}{rcl}\sqrt{56}&=&\sqrt{2^2 \cdot 14} \\ &=& \sqrt{2^2} \cdot \sqrt{14} \\ &=& 2 \sqrt{14} \\ \\\sqrt{108}&=&\sqrt{2^2 \cdot 3^2 \cdot 3} \\&=& \sqrt{2^2} \cdot \sqrt{3^2} \cdot \sqrt{3} \\ &=& 2 \cdot 3 \cdot \sqrt{3} \\ &=& 6 \sqrt{3} \\ \\ \dfrac{2}{\sqrt{5}}&=& \dfrac{2 \sqrt{5}}{\sqrt{5} \cdot \sqrt{5}}\\ &=& \dfrac{2 \sqrt{5}}{5} \\ \dfrac{3}{\sqrt{63}}&=&\dfrac{3}{\sqrt{3^2\cdot7}}\\&=& \dfrac{3}{3\cdot\sqrt{7}}\\&=&\dfrac{1}{\sqrt{7}}\\&=& \dfrac{\sqrt{7}}{\sqrt{7}\cdot\sqrt{7}}\\&=&\dfrac{\sqrt{7}}{7}\end{array}\]
#\begin{array}{rcl}
\sqrt{52}&=&\sqrt{4\times 13} \\ &&\phantom{xxx}\blue{52 \text{ written as a product inside the radical symbol}} \\
&=& \sqrt{{2}^2\times 13} \\ &&\phantom{xxx}\blue{4 \text{ written as a square}} \\
&=& \sqrt{{2}^2} \times \sqrt{13} \\ &&\phantom{xxx}\blue{\text{calculation rule: the root of a product is the product of roots}} \\
&=&2\sqrt{13} \\ &&\phantom{xxx}\blue{\text{root of a square is the absolute value of the number}} \end{array}#
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