Answer
False.
Work Step by Step
This is false. For instance, if we use $a=8$, $b=8$, then we get:
Left side: $\sqrt[3]{a+b}=\sqrt[3]{8+8}=\sqrt[3]{16}\approx 2.52$
Right side: $\sqrt[3]{a}+\sqrt[3]{b}=\sqrt[3]{8}+\sqrt[3]{8}=2+2=4$
And the sides do not equal.