First, note the order of operations: -3^2 means -(3^2) = -9. Then, cube it: (-9)^3 = -729. A negative number raised to an odd power remains negative.

Multiply the exponents: -2 * 2 = -4. This gives 2^-4. A negative exponent means taking the reciprocal, so 2^-4 = 1 / (2^4) = 1/16.

Multiply the exponents: 2 * 3 = 6. This gives 10^6. 10^6 is equal to 1 followed by 6 zeros, which is 1,000,000.

Multiply the powers: 1 * 4 = 4, resulting in 5^4. Evaluating 5^4 gives 5 * 5 * 5 * 5 = 625.

Applying the power rule of indices, you multiply the inner exponent by the outer exponent. 4 * 2 = 8, so the simplified expression is x^8.

Multiply the exponents together: 2 * 3 = 6. The expression becomes 3^6. Calculating 3^6 (3 * 3 * 3 * 3 * 3 * 3) gives 729.

According to the power of a power rule, multiply the exponents: 3 * 2 = 6. This gives 2^6. Expanding 2^6 yields 64.

This represents the general quotient rule of indices. When dividing terms with the same base 'a', the exponent of the denominator is subtracted from the exponent of the numerator, giving a^(x-y).

Subtract the powers: 4 - 3 = 1. This leaves 9^1. Any number raised to the power of 1 is the number itself, which is 9.

Applying the division rule of indices, 4^5 / 4^3 = 4^(5-3) = 4^2. Expanding this gives 4 * 4 = 16.