When multiplying terms with the same base, you add the exponents according to the product rule. Here, the base is 2, so 2^3 * 2^4 becomes 2^(3+4), which equals 2^7.

To simplify a base raised to a power and then raised to another power, multiply the two exponents together. Multiplying 2 and 4 gives 8, resulting in 3^8.

The rule for a power raised to a power requires multiplying the exponents. Calculating 2 times 6 gives 12, so the expression becomes a^12.

The principal square root of 64 is the positive number that yields 64 when squared. Since 8^2 is 64, sqrt(64) equals 8.

When dividing exponential terms with the same base, you subtract the exponent of the denominator from the exponent of the numerator. Here, 8 - 3 equals 5, resulting in 2^5.

By the rules of negative exponents, moving the base to the denominator makes the exponent positive. Thus, a^-5 is rewritten as 1/a^5.

Computing approximate decimal values: 9/31 ≈ 0.290, 3/17 ≈ 0.176, 6/23 ≈ 0.261, 4/11 ≈ 0.364, 7/25 = 0.280. The maximum value among these corresponds to 4/11.

Diving both structured sides securely by 1.125 yields the equation 10^k = 0.001. A value of 0.001 represents 10^-3, demonstrating effectively that the exponent k equals -3.

Evaluating the fractions as decimals: 11/17 ≈ 0.647, 7/11 ≈ 0.636, 5/9 ≈ 0.556, and 8/15 ≈ 0.533. The only sequence strictly decreasing from largest to smallest is provided in option D.

When raising a power to another power, you multiply the exponents. Multiplying the inside exponent 3 by the outside exponent 2 yields an exponent of 6, resulting in 2^6.