When calculating with exponents, a couple of rules hold.
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For integer #\orange n# and #\purple m# we have:
\[\blue a^\orange n \cdot \blue a^\purple m=\blue a^{\orange n+\purple m}\]
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Example
\[\begin{array}{rcl}\blue{x}^\orange 2 \cdot \blue{x}^\purple 4 &=&
\blue{x}^{\orange 2+\purple 4}\\ &=& \blue{x}^6\\ \end{array}\]
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We prove this for integer #\orange n \geq 0# and integer #\purple m \geq 0#:
\[\begin{array}{rcl}\blue a^\orange n \cdot \blue a^\purple m & =& \underbrace{\blue a \cdot \blue a \cdot \ldots \cdot \blue a}_{\orange n\textrm{ times}}\cdot\underbrace{\blue a \cdot \blue a \cdot \ldots \cdot \blue a}_{\purple m\textrm{ times}} \\ &=&\underbrace{\blue a \cdot \blue a \cdot \ldots \cdot \blue a}_{\orange n+\purple m\textrm{ times}}\\&=&{\blue a}^{\orange n+\purple m}\end{array}\]
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Example
\[\begin{array}{rcl}\blue{x}^\orange 2 \cdot \blue{x}^\purple 4 &=& \underbrace{\blue x \cdot \blue x}_{\orange 2\textrm{ times}}\cdot\underbrace{\blue x \cdot \blue x \cdot \blue x \cdot \blue x}_{\purple 4\textrm{ times}} \\ &=& \underbrace{\blue x \cdot \blue x \cdot \blue x \cdot \blue x \cdot \blue x \cdot \blue x}_{\orange 2+\purple 4\textrm{ times}}\\&=&
\blue{x}^{\orange 2+\purple 4}\\ \end{array}\]
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For integer #\orange n# and #\purple m# we have:
\[\frac{\blue{a}^{\orange{n}}}{\blue{a}^{\purple{m}}}=\blue{a}^{\orange{n}-\purple{m}}\]
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Example
\[\begin{array}{rcl} \dfrac{\blue{x}^{\orange{5}}}{\blue{x}^{\purple{3}}}&=&\blue{x}^{\orange{5}-\purple{3}}\\ &=& \blue{x}^2 \end{array}\]
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For integer #\orange n# we have:
\[\left(\frac{\blue a}{\green b}\right)^\orange n=\dfrac{\blue a^\orange n}{\green b^\orange n}\]
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Example
\[\begin{array}{rcl}\left(\dfrac{\blue{x}}{\green{y}}\right)^{\orange 3} &=& \dfrac{\blue{x}^\orange 3}{\green{y}^\orange 3} \end{array}\]
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We prove this for #\orange n\geq 0#:
\[\left(\frac{\blue a}{\green b}\right)^\orange n=\underbrace{\frac{\blue a}{\green b} \cdot \frac{\blue a}{\green b} \cdot \ldots \cdot \frac{\blue a}{\green b}}_{\orange n\textrm{ times}}=\dfrac{\blue a^\orange n}{\green b^\orange n}\tiny.\]
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Example
\[\begin{array}{c}\left(\dfrac{\blue{x}}{\green{y}}\right)^\orange 3 =
\underbrace{\dfrac{\blue{x}}{\green{y}} \cdot \dfrac{\blue{x}}{\green{y}} \cdot \dfrac{\blue{x}}{\green{y}}}_{\orange 3\textrm{ times}}= \dfrac{\blue{x}^\orange 3}{\green{y}^\orange3} \end{array}\]
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For integer #\orange n# we have:
\[\left({\blue a}\cdot{\green b}\right)^{\orange n}={\blue a^\orange n}\cdot{\green b^\orange n}\]
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Example
\[\begin{array}{rcl}\left(\blue{x}\cdot \green{y}\right)^{\orange 3} &=& \blue{x}^\orange 3 \cdot \green{y}^\orange3
\end{array}\]
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We prove this for #\orange n\geq 0#:
\[\begin{array}{rcl}\left({\blue a}\cdot{\green b}\right)^{\orange n}&=&\underbrace{(\blue a \cdot \green b) \cdot (\blue a \cdot \green b) \cdot \ldots \cdot (\blue a \cdot \green b)}_{\orange n\textrm{ times}} \\ &=& \underbrace{\blue a \cdot \blue a \cdot \ldots \cdot \blue a}_{\orange n\textrm{ times}} \cdot \underbrace{\green b \cdot \green b \cdot \ldots \cdot \green b}_{\orange n\textrm{ times}} \\&=& {\blue a^\orange n}\cdot{\green b^\orange n}\end{array}\]
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Example
\[\begin{array}{rcl}\left(\blue{x}\cdot \green{y}\right)^{\orange 3} &=& \underbrace{\blue{x} \cdot \green{y} \cdot \blue{x} \cdot \green{y}\cdot \blue{x} \cdot \green{y}}_{\orange 3\textrm{ times}} \\ &=& \underbrace{\blue x \cdot \blue x \cdot \blue x}_{\orange 3\textrm{ times}} \cdot \underbrace{\green y \cdot \green y \cdot \green y}_{\orange 3\textrm{ times}} \\&=& \blue{x}^\orange 3 \cdot \green{y}^\orange3
\end{array}\]
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For integer #\orange n# and #\purple m# we have:
\[\left({\blue a}^{\orange n}\right)^{\purple m}=\blue a^{\orange n\cdot \purple m}\]
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Example
\[\begin{array}{rcl}\left(\blue{x}^{\orange 2}\right)^{\purple 4} &=& \blue{x}^{\orange 2 \cdot \purple 4} \\ &=& \blue x^8 \end{array}\]
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We prove this for #\orange n \geq 0# and #\purple m \geq 0#:
\[\begin{array}{rcl}\left(\blue a^\orange n \right)^\purple m & =& \underbrace{\blue a^\orange n \cdot \blue a^\orange n \cdot \cdots \cdot \blue a^\orange n}_{\purple m\textrm{ times}} \\ &=&\underbrace{\blue a \cdot \blue a \cdot \cdots \cdot \blue a}_{\orange n\cdot \purple m\textrm{ times}}\\&=&\blue a^{\orange n\cdot \purple m}\end{array}\]
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Example
\[\begin{array}{rcl}\left(\blue{x}^{\orange2}\right)^{\purple 3} &=& \underbrace{\blue{x}^\orange2 \cdot \blue{x}^\orange2 \cdot \blue{x}^\orange2}_{\purple 3 \text{ times}} \\&=& \underbrace{\blue{x} \cdot \blue{x} \cdot \blue{x} \cdot \blue{x} \cdot \blue{x} \cdot \blue{x}}_{\orange 2 \cdot \purple 3 \text{ times}} \\ &=& \blue{x}^{\orange2 \cdot \purple3} \end{array}\]
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Eliminate the brackets and write as simple as possible: #\left(4 \cdot b^2\right)^{3}#.
#64\cdot b^6#
#\begin{array}{rcl}
\left(4 \cdot b^2\right)^{3} &=& 4^{3} \cdot \left(b^{2}\right)^{3} \\
&&\phantom{xx}\blue{\text{rule } \left(a \cdot b\right)^n = a^{n} \cdot b^{n}}\\
&=& 4^3 \cdot b^{2 \cdot 3} \\
&&\phantom{xx}\blue{\text{rule } \left(a^n\right)^m = a^{n \cdot m}} \\
&=& 4^{3} \cdot b^{6} \\
&&\phantom{xx}\blue{\text{multiplied in the exponent }} \\
&=& 64\cdot b^6 \\
&&\phantom{xx}\blue{4^3\text{ calculated }}\\
\end{array}#
Calculating with integer exponents
