\text { 2) } \frac{2 x^0 y^{-2}}{\left((2 z)^4 \cdot\left(2 x^3 y^2 z^{-3}\right)^0\right)^{-2}}

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Let's simplify this expression with exponents. We'll start by identifying terms with exponent zero. Remember that any number raised to the power of zero equals one. So x to the power of zero equals one, and the term 2 x cubed y squared z to the power of negative 3, all raised to the power of zero, also equals one. Now we can substitute these values into our original expression. This gives us 2 times 1 times y to the power of negative 2, divided by 2z to the power of 4, times 1, all raised to the power of negative 2. This simplifies to 2 y to the power of negative 2, divided by 2z to the power of 4, all raised to the power of negative 2. Now let's continue simplifying. In step 3, we rewrite the negative exponent in the numerator. Remember that y to the power of negative 2 equals 1 divided by y squared. This gives us 2 divided by y squared, over the denominator. In step 4, we simplify the denominator using the power of a power rule. When we have a power raised to another power, we multiply the exponents. So 2z to the power of 4, all raised to the power of negative 2, equals 2z to the power of 4 times negative 2, which is 2z to the power of negative 8. In step 5, we rewrite the negative exponent in the denominator. 2z to the power of negative 8 equals 1 divided by 2z to the power of 8. So our expression becomes 2 divided by y squared, divided by 1 divided by 2z to the power of 8. Now we're ready for the final steps. In step 6, we divide the fractions by multiplying the numerator by the reciprocal of the denominator. When we divide by a fraction, we multiply by its reciprocal. So 2 divided by y squared, divided by 1 divided by 2z to the power of 8, equals 2 divided by y squared, multiplied by 2z to the power of 8 divided by 1. This gives us 2 times 2z to the power of 8, all divided by y squared. In step 7, we simplify 2z to the power of 8 using the power of a product rule. When we have a product raised to a power, we raise each factor to that power. So 2z to the power of 8 equals 2 to the power of 8 times z to the power of 8, which is 256 times z to the power of 8. In our final step, we substitute this back into our expression and multiply the constants. 2 times 256 is 512, so our final answer is 512 z to the power of 8, divided by y squared. Let's summarize the key exponent rules we used to simplify this expression. First, any number raised to the power of zero equals one. This helped us eliminate terms in our original expression. Second, for negative exponents, x to the power of negative n equals 1 divided by x to the power of n. We used this rule to rewrite terms with negative exponents in both the numerator and denominator. Third, the power of a power rule states that x to the power of m, all raised to the power of n, equals x to the power of m times n. Fourth, the power of a product rule states that x times y, all raised to the power of n, equals x to the power of n times y to the power of n. Finally, when dividing fractions, we multiply by the reciprocal of the divisor. By applying these rules systematically, we transformed our complex expression into the simplified form: 512 z to the power of 8, divided by y squared.