If $$ 10^x = \frac{1}{2} $$, then $$ 10^{-8x} = $$ $x$

Concept & Formula This problem is solved using the "Power of a Power" rule in exponents, which allows us to manipulate the given equation to match the target expression. $$ (a^m)^n = a^{m \times n} $$ $$ a^{-m} = \frac{1}{a^m} $$ Step-by-Step Solution * We are given that $$ 10^x = \frac{1}{2} $$. * We need to find the value of $$ 10^{-8x} $$. We can rewrite this target expression using the power of a power rule: $$ 10^{-8x} = (10^x)^{-8} $$ * Now, substitute the given value of $$ 10^x $$ into our rewritten expression: $$ \left(\frac{1}{2}\right)^{-8} $$ * Apply the negative exponent rule to flip the fraction and make the exponent positive: $$ \left(\frac{2}{1}\right)^8 = 2^8 $$ * Calculate the final value: $$ 2^8 = 256 $$ Exam Strategy & Shortcut Instead of trying to solve for $x$ using logarithms (which wastes massive amounts of time), manipulate the base structure. You need $$-8x$$ in the exponent, so immediately raise both sides of the known equation to the power of $$-8$$. The right side becomes $$(1/2)^{-8}$$. Since flipping the fraction negates the exponent, this is simply $$2^8$$. Memorizing powers of 2 up to $$2^{10}$$ ($$1024$$) makes calculating $$2^8 = 256$$ an instantaneous mental step. Common Pitfall A frequent trap is ignoring the negative sign in the exponent and simply calculating $$(1/2)^8 = 1/256$$, leading students straight to option (a). Always remember that a negative exponent on a fraction effectively turns it upside down. Final Answer **Therefore, the correct answer is 256.**