Making use of the cube root table,

By prime factorisation, we have:

$1346=2 \times 673 \Rightarrow \sqrt[3]{1346}=\sqrt[3]{2} \times \sqrt[3]{673}$

Also

$670<673<680 \Rightarrow \sqrt[3]{670}<\sqrt[3]{673}<\sqrt[3]{680}$

From the cube root table, we have:

$\sqrt[3]{670}=8.750$ and $\sqrt[3]{680}=8.794$

For the difference (680-">−-670), i.e., 10, the difference in the values

$=8.794-8.750=0.044$

$\therefore$ For the difference of $(673-670)$, i.e., 3 , the difference in the values

$=\frac{0.044}{10} \times 3=0.0132=0.013$ (upto three decimal places)

$\therefore \sqrt[3]{673}=8.750+0.013=8.763$

Now

$\sqrt[3]{1346}=\sqrt[3]{2} \times \sqrt[3]{673}=1.260 \times 8.763=11.041$ (upto three decimal places)

Thus, the answer is 11.041.