First of all, we can split the root of the multiplication into the multiplication of the roots:
[tex]\sqrt[3]{-56ab^6c^{10}}=\sqrt[3]{-56}\cdot\sqrt[3]{a}\cdot\sqrt[3]{b^6}\cdot\sqrt[3]{c^10}[/tex]
If we look at the prime factorization of -56 we have
[tex]-56=2^3\cdot 7[/tex]
So, we have
[tex]\sqrt[3]{-56} = -\sqrt[3]{2^3\cdot 7} = -\sqrt[3]{2^3}\cdot\sqrt[3]{7} = -2\sqrt[3]{7}[/tex]
We can do nothing about [tex]\sqrt[3]{a}[/tex], because the exponent is lower than the order of the root.
We have
[tex]\sqrt[3]{b^6} = (b^6)^{\frac{1}{3}} = b^\frac{6}{3}=b^2[/tex]
Finally, we have
[tex]\sqrt[3]{c^{10}} =\sqrt[3]{c^{9+1}}=\sqrt[3]{c^{9}\cdot c}=c^\frac{9}{3}\cdot\sqrt[3]{c}=c^3\sqrt[3]{c}[/tex]
So, the whole expression is equivalent to
[tex]-2b^2c^3\sqrt[3]{7ac}[/tex]
