To simplify the expression \( \left( 4^{3 \times 7} \right)^{-5} \div \left( 7^{9 \times 4} \right)^{-13} \) using the properties of integer exponents, we'll follow these steps:
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Calculate the exponents:
\( 3 \times 7 = 21 \)
\( 9 \times 4 = 36 \)Therefore, we can rewrite the expression as: \[ \left( 4^{21} \right)^{-5} \div \left( 7^{36} \right)^{-13} \]
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Apply the negative exponent rule:
Recall that \( a^{-m} = \frac{1}{a^{m}} \). Thus, we can rewrite the expression: \[ \frac{1}{4^{21 \cdot 5}} \div \frac{1}{7^{36 \cdot 13}} \] This simplifies to: \[ \frac{7^{36 \cdot 13}}{4^{21 \cdot 5}} \] -
Calculate the new exponents from the multiplication:
\( 21 \cdot 5 = 105 \)
\( 36 \cdot 13 = 468 \)Now the expression is: \[ \frac{7^{468}}{4^{105}} \]
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Rewrite \( 4 \) in terms of \( 2 \):
Since \( 4 = 2^2 \), we can express \( 4^{105} \) as \( (2^2)^{105} = 2^{2 \cdot 105} = 2^{210} \).Therefore, we now have: \[ \frac{7^{468}}{2^{210}} \]
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Final expression:
Thus the equivalent expression with no remaining exponents and fully simplified is: \[ \frac{7^{468}}{2^{210}} \]
This fraction cannot be simplified further since 7 and 2 are prime bases. The final answer is: \[ \frac{7^{468}}{2^{210}} \]
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