$$ (64)^{-\frac{1}{2}} - (-32)^{-\frac{4}{5}} = $$ $x$

Concept & Formula This question combines the evaluation of fractional exponents with the application of negative bases. $$ (x^m)^{\frac{1}{n}} = (x^{\frac{1}{n}})^m $$ $$ a^{-n} = \frac{1}{a^n} $$ Step-by-Step Solution * Evaluate the first term: $$ (64)^{-\frac{1}{2}} $$ Since $$ 64 = 8^2 $$, apply the power rule: $$ (8^2)^{-\frac{1}{2}} = 8^{-1} = \frac{1}{8} $$ * Evaluate the second term: $$ (-32)^{-\frac{4}{5}} $$ Recognize that $$ -32 = (-2)^5 $$. Apply the power rule: $$ ((-2)^5)^{-\frac{4}{5}} = (-2)^{-4} $$ * Convert the negative exponent into a fraction: $$ (-2)^{-4} = \frac{1}{(-2)^4} $$ Since any negative number raised to an even power becomes positive: $$ (-2)^4 = 16 $$ So, the second term simplifies to $$ \frac{1}{16} $$. * Substitute both simplified terms back into the main expression: $$ \frac{1}{8} - \frac{1}{16} $$ * Find a common denominator (16) and subtract: $$ \frac{2}{16} - \frac{1}{16} = \frac{1}{16} $$ Exam Strategy & Shortcut Break down complex terms independently. For the second term, immediately note the denominator of the exponent is 5, meaning you need a 5th root. The 5th root of -32 is exactly -2. Then, raise -2 to the remaining numerator power, which is -4. So, it becomes $$(-2)^{-4}$$, yielding $$1/16$$. The first term is simply the reciprocal of the square root of 64, giving $$1/8$$. Simply calculate $$1/8 - 1/16 = 1/16$$ instantly. Common Pitfall The most dangerous pitfall here is mishandling the negative base in the second term. Students often see the negative sign in $$-32$$ and incorrectly carry it forward to make the final fraction negative, completely forgetting that raising a negative number to an even power (the 4 in the exponent numerator) makes the result positive. Final Answer **Therefore, the correct answer is $$ \frac{1}{16} $$.**