$$ \left(\frac{1}{216}\right)^{-\frac{2}{3}} \div \left(\frac{1}{27}\right)^{-\frac{4}{3}} = $$ $x$

Concept & Formula This problem tests the inversion rule for negative exponents applied to fractions, followed by exponent simplification. $$ \left(\frac{a}{b}\right)^{-n} = \left(\frac{b}{a}\right)^n $$ $$ (x^a)^b = x^{ab} $$ Step-by-Step Solution * First, eliminate the negative signs in the exponents by reciprocating the inner fractions: $$ \left(\frac{1}{216}\right)^{-\frac{2}{3}} = (216)^{\frac{2}{3}} $$ $$ \left(\frac{1}{27}\right)^{-\frac{4}{3}} = (27)^{\frac{4}{3}} $$ * Convert the bases into their prime power equivalents. $$ 216 = 6^3 $$ and $$ 27 = 3^3 $$. * Apply the power of a power rule: $$ (6^3)^{\frac{2}{3}} = 6^2 = 36 $$ $$ (3^3)^{\frac{4}{3}} = 3^4 = 81 $$ * Substitute these simplified values back into the original division operation: $$ 36 \div 81 = \frac{36}{81} $$ * Simplify the fraction by dividing the numerator and denominator by their greatest common divisor (9): $$ \frac{36 \div 9}{81 \div 9} = \frac{4}{9} $$ Exam Strategy & Shortcut Flip the fractions immediately to remove negative powers. Recognize that $$ 216 $$ and $$ 27 $$ are cubes. The denominator 3 in the fractional exponents signals that taking the cube root is your first step. Cube root of 216 is 6; square it to get 36. Cube root of 27 is 3; raise it to the 4th power to get 81. The ratio $$ 36/81 $$ visually reduces to $$ 4/9 $$ quickly using the 9 times table. Common Pitfall A major trap is attempting to perform the division operation before simplifying the individual exponential terms. This leads to convoluted algebraic expressions like $$ (1/216 \div 1/27) $$, which jumbles the differing exponents and nearly guarantees a calculation error. Always simplify bases independently first. Final Answer **Therefore, the correct answer is $$ \frac{4}{9} $$.**