To find an equivalent expression to the given scenarios using the Property of Negative Integer Exponents, we will first address each component in the expression.
We start with the expression:
\[ \frac{14^{7^3}}{25^{-4}} \]
Step 1: Simplifying the denominator with negative exponent
The denominator contains \(25^{-4}\). According to the Property of Negative Integer Exponents, we can rewrite this as:
\[ 25^{-4} = \frac{1}{25^4} \]
Thus, when we replace \(25^{-4}\) in our original fraction, we have:
\[ \frac{14^{7^3}}{25^{-4}} = 14^{7^3} \cdot 25^4 \]
Step 2: Simplifying the expression
Now the expression is:
\[ 14^{7^3} \cdot 25^4 \]
Since we need to ensure that we only use positive exponents, notice that both exponents are positive in their current forms.
Step 3: Calculating the power
Next, we need to calculate \(7^3\):
\[ 7^3 = 343 \]
Thus, we can substitute that back into our expression:
\[ 14^{343} \cdot 25^4 \]
Final Result:
The equivalent expression to the numerator \(14^{7^3}\) with the denominator \(25^{-4}\), where all exponents are positive, is:
\[ 14^{343} \cdot 25^4 \]
This is the expression in terms of positive exponents.
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