Answer: $3$

Explanation:1. Use the definition of the function $f(x)=c^x$: $f(7) = c^7$ $f(5) = c^5$

2. Substitute these into the given equation $f(7)=9 \cdot f(5)$: $c^7 = 9 \cdot c^5$

3. Solve for c: Since c is a constant greater than $1$ ($c > 1$), $c^5$ is not equal to $0$, so we can divide both sides of the equation by $c^5$: $\dfrac{c^7}{c^5} = 9$ Using the exponent rule $\dfrac{a^m}{a^n} = a^{m-n}$: $c^{7-5} = 9$ $c^2 = 9$

4. Find the value of c: Take the square root of both sides: $c = \pm\sqrt{9}$ $c = \pm3$ The problem states that c is a constant greater than $1$. Therefore, we choose the positive value. $c = 3$

Conclusion:The value of c is $3$.