Numbers: Powers and roots
Rules of calculation for powers
To make calculating with powers easier, we can use some rules of calculation.
Multiplication of powers
When we multiply #\blue2^\orange3# by #\blue2^\purple4#, we get:
\[\begin{array}{rcl}\blue2^\orange3 \times \blue2^\purple4&=&\underbrace{(\blue2 \times \blue2 \times \blue2)} _{\orange3 \text { times}}\times \underbrace{(\blue2 \times \blue2 \times \blue2 \times \blue2)}_{\purple4 \text{ times}}\\&=&\blue2^{\orange3+\purple4}\\&=&\blue2^7\end{array}\]
In general we can state the following:
When multiplying powers with the same #\blue{\textit{base}}#, we should add the exponents. The #\blue{\textit{base}}# remains the same.
Examples
\[\begin{array}{rcl}\blue3^\orange{4} \times \blue3^\purple{2} &=&\blue3^{\orange4+\purple2} \\&=&\blue3^{6} \\ \\ (\blue{\frac{1}{2}})^\orange{5} \times (\blue{\frac{1}{2}})^\purple3&=&(\blue{\frac{1}{2}})^{\orange5+\purple3} \\&=&(\blue{\frac{1}{2}})^{8} \\ \\ (\blue{-3})^\orange{10} \times (\blue{-3})^\purple{5}&=&(\blue{-3})^{\orange{10}+\purple5} \\&=&(\blue{-3})^{15} \end{array}\]
Division of powers
When we divide #\blue2^\orange{5}# by #\blue2^\purple2#, we get:
\[\begin{array}{rcl}\blue2^\orange5 : \blue2^\purple2&=&\underbrace{(\blue2 \times \blue2 \times \blue2 \times \blue2 \times \blue2)}_{\orange5 \text{ times}} : \underbrace{(\blue2 \times \blue2)}_{\purple2 \text{ times}}\\&=&\underbrace{\blue2 \times \blue2 \times \blue2}_{\orange5-\purple2 \text { times}}\\&=&\blue2^{\orange5-\purple2}\\&=&\blue2^3\end{array}\]
In general we can state the following:
When dividing powers with the same #\blue{\textit{base}}#, we should subtract the exponents. The #\blue{\textit{base}}# remains the same.
Examples
\[\begin{array}{rcl}\blue3^\orange{4} : \blue3^\purple{2}&=&\blue3^{\orange4-\purple2} \\&=&\blue3^{2} \\ \\ (\blue{\frac{1}{2}})^\orange{5} :(\blue{\frac{1}{2}})^\purple3&=&(\blue{\frac{1}{2}})^{\orange5-\purple3} \\&=&(\blue{\frac{1}{2}})^{2} \\ \\ (\blue{-3})^\orange{10} : (\blue{-3})^\purple{5}&=&(\blue{-3})^{\orange{10}-\purple5} \\&=&(\blue{-3})^{5}\end{array}\]
Exponentiation of powers
When we raise #\blue2^\orange{3}# to the power #\purple4#, we get:
\[\begin{array}{rcl}\left(\blue2^\orange3\right)^\purple4&=&\underbrace{\blue2^\orange3 \times \blue2^\orange3 \times \blue2^\orange3 \times \blue2^\orange3}_{\purple4 \text { times}}\\&=&\blue2^{\overbrace{\orange3+\orange3+\orange3+\orange3}^{\purple4 \text{ times}}}\\&=&\blue2^{\orange3 \times \purple4}\\&=&\blue2^{12}\end{array}\]
In general we can state the following:
When raising a power to a power, we should multiply the exponents. The #\blue{\textit{base}}# remains the same.
Example
\[\begin{array}{rcl}\left(\blue3^\orange{4}\right)^\purple{2} &=& \blue3^{\orange4 \times \purple2} \\ &=& \blue3^{8} \\ \\ \left(\left(\blue{\frac{1}{2}}\right)^\orange{5}\right)^\purple3&=&(\blue{\frac{1}{2}})^{\orange5 \times \purple3} \\&=&(\blue{\frac{1}{2}})^{15} \\ \\ \left(\left(\blue{-3}\right)^\orange{10}\right)^\purple{5} &=& (\blue{-3})^{\orange{10} \times \purple5} \\&=&(\blue{-3})^{50} \end{array}\]
#\begin{array}{rcl}
12^4 \times 12^2 &=& 12^{4+2} \\
&&\phantom{xxx}\blue{\text{calculation rule: when multiplying powers with the same base, }}\\
&&\phantom{xxx}\blue{\text{we add the exponents}}\\
&=& 12^{6}\\
& &\phantom{xxx}\blue{\text{added exponents}}\\
\end{array}#
Therefore, the number #6# should be entered on the dots.
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