1. Problem: Simplify $y^5 \cdot y^7$. Step 1: Use the product of powers rule: $a^m \cdot a^n = a^{m+n}$. Step 2: Apply the rule: $y^5 \cdot y^7 = y^{5+7} = y^{12}$. Step 3: The circled answer is $y^{11}$, which is incorrect. 2. Problem: Simplify $(y^5)^2$. Step 1: Use the power of a power rule: $(a^m)^n = a^{m \cdot n}$. Step 2: Apply the rule: $(y^5)^2 = y^{5 \cdot 2} = y^{10}$. Step 3: The circled answer is $y^{10}$, which is correct. 3. Problem: Simplify $\frac{m^8}{m^3}$. Step 1: Use the quotient of powers rule: $\frac{a^m}{a^n} = a^{m-n}$. Step 2: Apply the rule: $\frac{m^8}{m^3} = m^{8-3} = m^5$. Step 3: The circled answer is $m^5$, which is correct. 4. Problem: Simplify $(3x^2y^3)^2$. Step 1: Apply the power of a product rule: $(abc)^n = a^n b^n c^n$. Step 2: Expand: $(3x^2y^3)^2 = 3^2 \cdot (x^2)^2 \cdot (y^3)^2 = 9x^4y^6$. Step 3: The user states the answer is $\frac{5}{x^2}$. Step 4: The circled answer is 5, which is incorrect. 5. Problem: Simplify $4(x^{-3} x^5 y^{-1})^{-1}$. Step 1: Simplify inside the parentheses: $x^{-3} x^5 y^{-1} = x^{-3+5} y^{-1} = x^2 y^{-1}$. Step 2: Apply the negative exponent: $(x^2 y^{-1})^{-1} = x^{-2} y^{1}$. Step 3: Multiply by 4: $4 \cdot x^{-2} y = \frac{4y}{x^2}$. Step 4: The circled answer is $\frac{2x^{-2}y}{4}$ which simplifies to $\frac{x^{-2} y}{2} = \frac{y}{2x^2}$, which is incorrect compared to the correct $\frac{4y}{x^2}$. Summary: - Question 1: Wrong (circled $y^{11}$, correct $y^{12}$). - Question 2: Correct. - Question 3: Correct. - Question 4: Wrong (circled 5, correct $9x^4y^6$; user answer $\frac{5}{x^2}$ is incorrect). - Question 5: Wrong (circled answer incorrect, correct $\frac{4y}{x^2}$).