To find the missing exponent, we start with the equation given:
\[ \frac{5^{11} \cdot 5^?}{5^4 \cdot 5^?} = 5^{11 - 4} \]
First, simplify both sides of the equation.
On the left side: \[ 5^{11 + ?} / 5^{4 + ?} = 5^{(11 + ?) - (4 + ?)} = 5^{11 + ? - 4 - ?} = 5^{11 - 4} = 5^7 \]
Now we have: \[ 5^{11 - 4} = 5^7 \] To match exponents: \[ 11 + ? - (4 + ?) = 7 \]
This simplifies to: \[ 11 + ? - 4 - ? = 7 \] \[ 7 = 7 \]
This means that the equation holds true for any value of the missing exponent. To find the specific value of "?," we realize that "?," when isolated, could equal zero since it cancels itself out. Therefore, the missing exponent is:
\[ \boxed{0} \]
1 answer