The value of ${\left[5{\left(8^{\frac{1}{3}}+27^{\frac{1}{3}}\right)}^3\right]}^{\frac{1}{4}}$ is

Simplify inside the innermost parentheses first:
$$8^{\frac{1}{3}}=(2^3{)}^{\frac{1}{3}}=2$$
$$27^{\frac{1}{3}}=(3^3{)}^{\frac{1}{3}}=3$$
Substitute these values back into the expression:
$${\left[5(2+3{)}^3\right]}^{\frac{1}{4}}$$
$${\left[5(5{)}^3\right]}^{\frac{1}{4}}$$
$${\left[5^1\times 5^3\right]}^{\frac{1}{4}}={\left[5^4\right]}^{\frac{1}{4}}$$
$$5^{4\times \frac{1}{4}}=5^1=5$$