To simplify the expression \((12^{53}) / (7^{53}) \cdot (12^{106}) / (5^{106}) \cdot (5^{53})\), we can use the properties of exponents and specifically focus on the Quotient Rule and Product Rule.
- Using the Quotient Rule of Integer Exponents: \[ \frac{a^m}{b^m} = \left(\frac{a}{b}\right)^m \] Therefore, \[ \frac{12^{53}}{7^{53}} = \left(\frac{12}{7}\right)^{53} \]
So the expression simplifies to: \[ \left(\frac{12}{7}\right)^{53} \cdot \frac{12^{106}}{5^{106}} \cdot 5^{53} \]
- Now simplifying \( \frac{12^{106}}{5^{106}} \cdot 5^{53} \): \[ \frac{12^{106}}{5^{106}} \cdot 5^{53} = \frac{12^{106} \cdot 5^{53}}{5^{106}} = 12^{106} \cdot 5^{53 - 106} = 12^{106} \cdot 5^{-53} \]
Using the Property of exponents: \[ 5^{-53} = \frac{1}{5^{53}} \]
Thus, \[ 12^{106} \cdot 5^{-53} = \frac{12^{106}}{5^{53}} \]
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Putting it all together: The complete expression now looks like this: \[ \left(\frac{12}{7}\right)^{53} \cdot \frac{12^{106}}{5^{53}} = \frac{12^{53} \cdot 12^{106}}{7^{53} \cdot 5^{53}} \]
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Using the Product Rule: You can combine \(12^{53}\) and \(12^{106}\) as follows: \[ 12^{53} \cdot 12^{106} = 12^{53 + 106} = 12^{159} \]
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Finally, we have: \[ \frac{12^{159}}{7^{53} \cdot 5^{53}} = \frac{12^{159}}{(7 \cdot 5)^{53}} = \frac{12^{159}}{35^{53}} \]
This is the simplified form of the original expression: \[ \frac{12^{159}}{35^{53}} \]