If $a$ is a rational number and $b$ is an irrational number such that $ab$ is rational, then which of the following is false?

(A)$ab^2$ is irrational
(B)$a^2b$ is rational
(C)$\sqrt{ab}$ is rational
(D)$a+b$ is irrational
To determine which statement is false, we need to analyze the given conditions and each of the options provided.

Given:
- \( a \) is a rational number.
- \( b \) is an irrational number.
- \( ab \) is rational.

Let's analyze each option:

1. **Option (A): \( ab^2 \) is irrational**

   Since \( b \) is irrational, \( b^2 \) is also irrational (the product of an irrational number with itself is irrational). Therefore, \( ab^2 \) is the product of a rational number \( a \) and an irrational number \( b^2 \), which must be irrational. Hence, this statement is true.

2. **Option (B): \( a^2b \) is rational**

   Since \( a \) is rational, \( a^2 \) is also rational (the product of a rational number with itself is rational). Therefore, \( a^2b \) is the product of a rational number \( a^2 \) and an irrational number \( b \), which must be irrational. Hence, this statement is false.

3. **Option (C): \( \sqrt{ab} \) is rational**

   Since \( ab \) is rational (given), let \( ab = c \) where \( c \) is a rational number. The square root of a rational number is not necessarily rational. For example, \( \sqrt{2} \) is irrational. Therefore, this statement can be either true or false depending on the specific values of \( a \) and \( b \). Hence, this statement is not definitively false.

4. **Option (D): \( a + b \) is irrational**

   Since \( a \) is rational and \( b \) is irrational, the sum \( a + b \) is irrational (the sum of a rational number and an irrational number is irrational). Hence, this statement is true.

From the analysis, the false statement is:

The final answer is \( \boxed{\textbf{(B)}} \)