Answer
$2$
Work Step by Step
Simplify. $\sqrt[3]{\sqrt{\sqrt{169}+\sqrt {9}}+ \sqrt {\sqrt [3] {1000}+\sqrt[3] {216}}}$
Now, $\sqrt[3]{\sqrt{\sqrt{169}+\sqrt {9}}+ \sqrt {\sqrt [3] {1000}+\sqrt[3] {216}}}=\sqrt[3]{\sqrt{13+3}+ \sqrt {10+6}}$
or, $=\sqrt[3]{\sqrt{16}+ \sqrt {16}}$
or, $=\sqrt[3]{4+4}$
or, $=\sqrt[3]{8}$
or, $=\sqrt[3]{2^3}$
Hence, $\sqrt[3]{\sqrt{\sqrt{169}+\sqrt {9}}+ \sqrt {\sqrt [3] {1000}+\sqrt[3] {216}}}=2$