To simplify the expression \( (x^{22})(x^{7})^3 \), we will use the laws of exponents.
First, simplify \( (x^{7})^3 \): \[ (x^{7})^3 = x^{7 \cdot 3} = x^{21} \]
Now we substitute this back into the original expression: \[ (x^{22})(x^{21}) = x^{22 + 21} = x^{43} \]
Thus, we can see that \( p = 43 \).
Therefore, the value of \( p \) is \( \boxed{43} \).