To apply the property of negative integer exponents, we use the rule that states \( a^{-n} = \frac{1}{a^n} \).
Let's apply this to the expression \( \left( \frac{5}{4} \right)^{-9} \cdot 4^{-12} \).
First, we can rewrite each term with positive exponents:
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For \( \left( \frac{5}{4} \right)^{-9} \): \[ \left( \frac{5}{4} \right)^{-9} = \frac{1}{\left( \frac{5}{4} \right)^{9}} = \frac{1}{\frac{5^9}{4^9}} = \frac{4^9}{5^9} \]
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For \( 4^{-12} \): \[ 4^{-12} = \frac{1}{4^{12}} \]
Now, we can combine them: \[ \frac{4^9}{5^9} \cdot \frac{1}{4^{12}} = \frac{4^9}{5^9 \cdot 4^{12}} \]
Using the property of exponents \( a^m \cdot a^n = a^{m+n} \), we can simplify the \( 4^{9} \) and \( 4^{12} \): \[ 4^9 \cdot 4^{-12} = 4^{9-12} = 4^{-3} \] Thus, the combined expression becomes: \[ \frac{4^{-3}}{5^9} = \frac{1}{4^3 \cdot 5^9} \]
Finally, we rewrite the expression to ensure all exponents are positive: \[ \frac{1}{4^3 \cdot 5^9} \]
Hence, the equivalent expression with only positive exponents is: \[ \frac{1}{4^3 \cdot 5^9} \]