The value of $$ (8^{-25} - 8^{-26}) $$ is

Concept & Logic The problem tests your ability to factor out common terms with negative exponents. When dealing with negative exponents, the term with the larger absolute value in the exponent is actually the smaller quantity. We factor out the smaller term. $$ a^{-m} - a^{-(m+1)} = a^{-(m+1)} (a^1 - 1) $$ Step-by-Step Solution * Identify the smaller term. Between $$ 8^{-25} $$ and $$ 8^{-26} $$, $$ 8^{-26} $$ is the smaller value. * Rewrite the larger term ($$ 8^{-25} $$) in terms of the smaller term: $$ 8^{-25} = 8^{-26} \times 8^1 $$ * Substitute this back into the original expression: $$ (8^{-26} \times 8^1) - 8^{-26} $$ * Factor out the common term, $$ 8^{-26} $$: $$ 8^{-26}(8^1 - 1) $$ * Simplify the expression inside the parentheses: $$ 8 - 1 = 7 $$ * Combine the terms for the final factored form: $$ 7 \times 8^{-26} $$ Exam Strategy & Shortcut When subtracting powers of the same base that are exactly 1 unit apart (like -25 and -26), you can instantly apply a shortcut rule: factor out the smaller power and multiply by (base - 1). Here, the base is 8. The smaller power is $$ 8^{-26} $$. Multiply by $$ (8 - 1) = 7 $$. The answer is instantly $$ 7 \times 8^{-26} $$. This takes 5 seconds mentally. Common Pitfall A very common mistake is factoring out $$ 8^{-25} $$ instead of $$ 8^{-26} $$, which leaves a negative exponent inside the parenthesis, creating unnecessary fractions and confusion. Always factor out the lowest mathematical value, which is the "most negative" exponent. Final Answer **Therefore, the correct answer is $$ 7 \times 8^{-26} $$.**