1. **Problem:** If $a$ and $b$ are positive even integers, which of the following is greatest? Options: (a) $(-2a)^b$ (b) $(-2a)^{2b}$ (c) $(2a)^b$ (d) $2a^{2b}$ 2. **Formula and rules:** Since $a$ and $b$ are positive even integers, powers of negative numbers raised to even powers become positive. Also, compare the magnitude of each expression. 3. **Step-by-step evaluation:** - (a) $(-2a)^b = (-(2a))^b$. Since $b$ is even, $(-2a)^b = (2a)^b$. - (b) $(-2a)^{2b} = ((-2a)^2)^b = (4a^2)^b = 4^b a^{2b}$. - (c) $(2a)^b$ is the same as (a) after simplification. - (d) $2a^{2b}$ means $2 imes a^{2b}$. 4. **Compare magnitudes:** - (a) and (c) are equal: $(2a)^b$. - (b) is $4^b a^{2b} = (2^2)^b a^{2b} = 2^{2b} a^{2b} = (2^b a^b)^2 = ((2a)^b)^2$. - (d) is $2 a^{2b}$. Since $a,b$ are positive even integers, $(2a)^b$ is positive and $(2a)^{2b} = ((2a)^b)^2$ is the square of $(2a)^b$, so (b) is greater than (a) and (c). Also, (b) grows faster than (d) because $4^b a^{2b}$ grows faster than $2 a^{2b}$. 5. **Conclusion:** The greatest is option (b) $(-2a)^{2b}$. **Final answer:** (b) $(-2a)^{2b}$