Expression Comparison: Determining Greater Value When c>1

To solve this problem, we'll follow these steps:

• Simplify each expression using the rules of exponents.
• Compare the resulting powers since $c > 1$ implies that the expression with the highest power excites the largest value.

Now, let's work through each expression:
1. For choice (1) $c^2 \times c^{-3}$:
Applying the exponent rule $c^{2-3} = c^{-1}$.

2. For choice (2) $c^2 \times c^1$:
Using the exponent rule $c^{2+1} = c^3$.

3. For choice (3) $c^{-2} \times c^{-2}$:
Using the exponent rule $c^{-2-2} = c^{-4}$.

4. For choice (4) $(c)^4 \times c^1$:
Applying the exponent rule $c^{4+1} = c^5$.

Given that $c > 1$, the expression with the largest exponent will have the largest value. Comparing all the simplified expressions, we have exponents: $-1, 3, -4,$ and $5$.
Therefore, the expression with the greatest value is $(c)^4 \times c^1$, which corresponds to choice (4) since $c^5$ has the largest exponent.