1. Problem: Determine if the statement $(3x^2)^3 = 9x^6$ is TRUE or FALSE. Step 1: Use the power of a product rule: $(ab)^m = a^m b^m$. Step 2: Apply the rule: $(3x^2)^3 = 3^3 (x^2)^3 = 27 x^{6}$. Step 3: Compare with $9x^6$. Since $27x^6 \neq 9x^6$, the statement is FALSE. 2. Problem: Determine if $4a^{-2} \cdot 3a^3 = 12a$ is TRUE or FALSE. Step 1: Multiply coefficients: $4 \cdot 3 = 12$. Step 2: Use the product of powers rule: $a^{-2} \cdot a^3 = a^{-2+3} = a^{1} = a$. Step 3: So, $4a^{-2} \cdot 3a^3 = 12a$, which matches the statement. TRUE. 3. Problem: Determine if $(4xy^{-2})^{-2} = 4x^{-2} y^{4}$ is TRUE or FALSE. Step 1: Apply power of a product: $(4xy^{-2})^{-2} = 4^{-2} x^{-2} (y^{-2})^{-2}$. Step 2: Simplify powers: $4^{-2} = \frac{1}{16}$, $(y^{-2})^{-2} = y^{4}$. Step 3: So, $(4xy^{-2})^{-2} = \frac{1}{16} x^{-2} y^{4}$. Step 4: Compare with $4x^{-2} y^{4}$. They are not equal, so FALSE. 4. Problem: Determine if $-(-x)^3 = x^3$ is TRUE or FALSE. Step 1: Calculate $(-x)^3 = -x^3$ because cube preserves sign. Step 2: Then $-(-x)^3 = -(-x^3) = x^3$. Step 3: Statement is TRUE. 5. Problem: Determine if $(2^{-1} x^{2})^{3} \cdot (2^{-2} x^{-1})^{-2} = 2 x^{9}$ is TRUE or FALSE. Step 1: Simplify each term: $(2^{-1} x^{2})^{3} = 2^{-3} x^{6} = \frac{1}{8} x^{6}$. $(2^{-2} x^{-1})^{-2} = 2^{4} x^{2} = 16 x^{2}$. Step 2: Multiply: $\frac{1}{8} x^{6} \cdot 16 x^{2} = 2 x^{8}$. Step 3: Compare with $2 x^{9}$. Not equal, so FALSE. 6. Problem: Determine if $(ab)^m = a^m b^m$ is TRUE or FALSE. Step 1: This is the power of a product rule, which is TRUE. 7. Problem: Determine if $x^a \cdot x^b = x^{ab}$ is TRUE or FALSE. Step 1: The product of powers rule states $x^a \cdot x^b = x^{a+b}$, not $x^{ab}$. Step 2: So the statement is FALSE. 8. Problem: Determine if $(x^a y)^m \cdot (x^a y)^n = (x^a y)^{m+n}$ is TRUE or FALSE. Step 1: Use product of powers: $A^m \cdot A^n = A^{m+n}$. Step 2: Here $A = x^a y$, so the statement is TRUE. Final answers: 1. FALSE 2. TRUE 3. FALSE 4. TRUE 5. FALSE 6. TRUE 7. FALSE 8. TRUE