Under the given conditions, the simplified expression is

Given the conditions a > 0 and b < 0, let's simplify the expression
![\[ |ab^3| \]](https://img.qammunity.org/2024/formulas/mathematics/high-school/hxsg67epdnvmbdx4gdwfxhr8au1acca8hi.png)
Since a > 0, the absolute value of a is simply a.
![\[ a|b^3| \]](https://img.qammunity.org/2024/formulas/mathematics/high-school/m7x3gj4epcaauxajmwu6y9ewwsstxm7twh.png)
Since b < 0, the absolute value of b is -b.
![\[ a(-b^3) \]](https://img.qammunity.org/2024/formulas/mathematics/high-school/7gjne02xpvbft83k7oezd549x30eshdf81.png)
Now, we can simplify further:
![\[ -ab^3 \]](https://img.qammunity.org/2024/formulas/mathematics/high-school/y0gayaw7s7gt3ypfy506hrfmv6e7hlsbmx.png)
Therefore, under the given conditions, the simplified expression is
