1. The problem asks to write the product $xy \cdot xy \cdot xy \cdot xy$ using exponents. 2. Recall the rule for exponents: when multiplying the same base, add the exponents. For example, $a^m \cdot a^n = a^{m+n}$. 3. Here, the base is the product $xy$, and it is multiplied 4 times. 4. So, the product can be written as $\left(xy\right)^4$. 5. This means both $x$ and $y$ are each raised to the 4th power inside the parentheses. 6. Therefore, the expression is $\boxed{\left(xy\right)^4}$, which is equivalent to $x^4 y^4$ by the power of a product rule.