To solve for the missing exponent in the equation \( e^? \cdot e^{65} = e^{38} \), we can use the property of exponents that states \( e^a \cdot e^b = e^{a+b} \).
Let's denote the missing exponent as \( x \). The equation can be rewritten as:
\[ e^x \cdot e^{65} = e^{38} \]
Using the property of exponents, we combine the left side:
\[ e^{x + 65} = e^{38} \]
Since the bases are the same, we can set the exponents equal to each other:
\[ x + 65 = 38 \]
Now, we can solve for \( x \):
\[ x = 38 - 65 \] \[ x = -27 \]
Thus, the missing exponent is \( -27 \).
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