Exponents

${ 8 }^{ a+b }\quad =\quad 4\sqrt { 2 } \quad \quad \quad \\ \Rightarrow \quad { 2 }^{ 3a+3b }\quad =\quad { 2 }^{ 5/2 }\\ \Rightarrow \quad 3a+3b\quad =\quad \frac { 5 }{ 2 } \\ \Rightarrow \quad 6a+6b\quad =\quad 5\\ \Rightarrow \quad a+b\quad =\quad \frac { 5 }{ 6 } \quad ............{ eq }(1)\\ \\ \\ Similarly,\\ \\ { 8 }^{ b+c }\quad =\quad \sqrt { \frac { 2 }{ 16 } } \\ \Rightarrow \quad { 2 }^{ 3b+3c }\quad =\quad { 2 }^{ -3/2 }\\ \Rightarrow \quad 6b+6c\quad =\quad -3\quad ............eq(2)\\ \\ \\ Similarly,\\ \\ { 8 }^{ a+c }\quad =\quad 8\\ \Rightarrow \quad a+c\quad =\quad 1\\ \Rightarrow \quad c\quad =\quad 1-a\quad ................eq(3)\\ \\ \\ From\quad eq(2)\quad \& \quad eq(3),\quad we\quad get:\\ \\ a-b\quad =\quad \frac { 3 }{ 2 } ..................eq(4)\\ \\ By\quad Subtracting\quad eq(1)\quad from\quad eq(4),\quad we\quad get:\\ \\ -2b\quad =\quad \frac { 4 }{ 6 } \\ \Rightarrow \quad b\quad =\quad -\frac { 4 }{ 6\times 2 } \\ \Rightarrow \quad b\quad =\quad -\frac { 1 }{ 3 } \\ \therefore \quad b\quad =\quad -0.33\\ \\ \\$

(1) $\times$ (2) / (3): $8^{2b} = \frac{1}{4}, \\ 8^b = \frac{1}{2}, \\ b = -\frac{1}{3}.$