By the zero exponent rule, any non-zero value raised to 0 equals 1. Thus, x^0 = 1 and y^0 = 1. Adding them gives 1 + 1 = 2.

Flip the fraction to make the exponent positive: (2/3)^-3 = (3/2)^3. Then cube both the numerator and denominator: 3^3 / 2^3 = 27/8.

A negative exponent applied to a fraction flips the fraction (takes its reciprocal) and turns the exponent positive. So, (1/2)^-2 = (2/1)^2 = 2^2 = 4.

Apply the negative power rule: 3^-4 = 1 / (3^4). Expanding the denominator gives 3 * 3 * 3 * 3 = 81, so the result is 1/81.

Using the negative index rule, 10^-2 = 1 / (10^2) = 1/100. In decimal form, 1/100 is written as 0.01.

A negative exponent indicates the reciprocal of the base raised to the positive opposite exponent. So, 2^-3 = 1 / (2^3) = 1/8.

According to the zero index law, any non-zero number raised to the power of 0 is exactly 1. Therefore, 5^0 = 1.

This demonstrates the general power of a power rule for indices. The inner exponent 'a' is multiplied by the outer exponent 'b', resulting in y^(ab).

When raising a power to a power, you multiply the exponents. Here, -3 * -2 = 6. The negatives cancel out, leaving x^6.

Multiply the exponents: 2 * (1/2) = 1. This simplifies the expression to 4^1, which is simply 4.