The value of $$ \frac{1}{(216)^{-\frac{2}{3}}} + \frac{1}{(256)^{-\frac{3}{4}}} + \frac{1}{(32)^{-\frac{1}{5}}} $$ is

Concept & Formula The problem tests the laws of negative exponents and fractional indices. $$ a^{-m} = \frac{1}{a^m} \implies \frac{1}{a^{-m}} = a^m $$ $$ (x^a)^b = x^{ab} $$ Step-by-Step Solution * First term: $$ \frac{1}{(216)^{-\frac{2}{3}}} = (216)^{\frac{2}{3}} $$. Since $$ 216 = 6^3 $$, we apply the power rule to get $$ (6^3)^{\frac{2}{3}} = 6^2 = 36 $$. * Second term: $$ \frac{1}{(256)^{-\frac{3}{4}}} = (256)^{\frac{3}{4}} $$. Since $$ 256 = 4^4 $$, we get $$ (4^4)^{\frac{3}{4}} = 4^3 = 64 $$. * Third term: $$ \frac{1}{(32)^{-\frac{1}{5}}} = (32)^{\frac{1}{5}} $$. Since $$ 32 = 2^5 $$, we get $$ (2^5)^{\frac{1}{5}} = 2^1 = 2 $$. * Add the simplified terms together: $$ 36 + 64 + 2 = 102 $$. Exam Strategy & Shortcut Recognize perfect powers immediately. The numbers 216, 256, and 32 are common powers ($$ 6^3 $$, $$ 4^4 $$, and $$ 2^5 $$). The denominators of the given fractional exponents (3, 4, and 5) match these powers perfectly, meaning the bases will simplify to integers. A quick mental calculation of $$ 6^2 + 4^3 + 2^1 $$ yields 102 in seconds without writing anything down. Common Pitfall Students often mishandle the double negative when moving a term from the denominator to the numerator, mistakenly keeping the fractional exponent negative and getting stuck with complex fractions. Remember that flipping the position vertically strictly flips the exponent's sign. Final Answer **Therefore, the correct answer is 102.**