This tests applying integer exponent properties: product rule ($a^m \times a^n = a^{m+n}$), quotient rule ($a^m \div a^n = a^{m-n}$), power rule ($(a^m)^n = a^{m n}$), zero exponent ($a^0 = 1$), and negative exponents ($a^{-n} = \frac{1}{a^n}$). For expression $3^4 \times 3^{-2}$, apply product rule: same base 3, add exponents $4 + (-2) = 2$, giving $3^2$, then evaluate: $3^2 = 9$. For $(2^3)^4$, multiply exponents: $3 \times 4 = 12$ giving $2^{12}$, evaluating: $2^{12} = 4096$. Key principle: properties apply when bases are identical; different bases ($2^3 \times 3^2$) can't combine using these rules. For ($4^{-3}$), apply the negative exponent rule: ($4^{-3} = \frac{1}{4^3} = \frac{1}{64}$). Choice D is correct because it correctly evaluates $a^{-n} = \frac{1}{a^n}$ by taking the reciprocal and raising to the positive power. Choice A ignores the reciprocal and adds a negative sign unnecessarily. Strategy: (1) identify operation (multiply/divide/power of power), (2) check bases match (must be same for properties), (3) apply correct rule (add/subtract/multiply exponents), (4) simplify further (combine like terms, evaluate if needed), (5) handle special cases (zero $\to$ 1, negative $\to$ reciprocal). Common mistakes: multiplying bases when should add exponents, confusing rules (adding when should multiply for powers), working with different bases ($2^3 \times 3^2$ can't simplify using exponent rules).

This tests applying integer exponent properties: product rule ($a^m \times a^n = a^{m+n}$), quotient rule ($a^m \div a^n = a^{m-n}$), power rule ($(a^m)^n = a^{m n}$), zero exponent ($a^0 = 1$), and negative exponents ($a^{-n} = \frac{1}{a^n}$). For expression $3^4 \times 3^{-2}$, apply product rule: same base 3, add exponents $4 + (-2) = 2$, giving $3^2$, then evaluate: $3^2 = 9$. For $(2^3)^4$, multiply exponents: $3 \times 4 = 12$ giving $2^{12}$, evaluating: $2^{12} = 4096$. Key principle: properties apply when bases are identical; different bases ($2^3 \times 3^2$) can't combine using these rules. For $(\left(3^2 \times 3^{-5}\right) \div 3^{-1})$, first apply the product rule inside the parentheses: $2 + (-5) = -3$, giving $(3^{-3})$, then use the quotient rule: $-3 - (-1) = -2$, resulting in $(3^{-2})$. Choice B is correct because it properly applies the product rule by adding exponents and then the quotient rule by subtracting exponents, correctly handling the negative exponent. Choice A ignores negative making $3^{-3} = 3^3$ when should be $1/27$ or confuses subtraction in quotient. Strategy: (1) identify operation (multiply/divide/power of power), (2) check bases match (must be same for properties), (3) apply correct rule (add/subtract/multiply exponents), (4) simplify further (combine like terms, evaluate if needed), (5) handle special cases (zero → 1, negative → reciprocal). Common mistakes: multiplying bases when should add exponents, confusing rules (adding when should multiply for powers), working with different bases ($2^3 \times 3^2$ can't simplify using exponent rules).

This tests applying integer exponent properties: product rule (aᵐ × aⁿ = aᵐ⁺ⁿ), quotient rule (aᵐ ÷ aⁿ = aᵐ⁻ⁿ), power rule ((aᵐ)ⁿ = aᵐⁿ), zero exponent (a⁰=1), and negative exponents (a⁻ⁿ=1/aⁿ). For expression 3⁴ × 3⁻², apply product rule: same base 3, add exponents 4+(-2)=2, giving 3², then evaluate: 3²=9. For (2³)⁴, multiply exponents: 3×4=12 giving 2¹², evaluating: 2¹²=4096. Key principle: properties apply when bases are identical; different bases (2³ × 3²) can't combine using these rules. For $(\left(2^3$$\right)^4$ \div $2^5$), first apply the power rule to get $(2^{12}$), then use the quotient rule to subtract exponents: 12 - 5 = 7, resulting in $(2^7$). Choice B is correct because it properly applies the power rule by multiplying exponents and then the quotient rule by subtracting exponents. Choice A multiplies bases instead of adding exponents or confuses the rules by adding when should multiply for power of power. Strategy: (1) identify operation (multiply/divide/power of power), (2) check bases match (must be same for properties), (3) apply correct rule (add/subtract/multiply exponents), (4) simplify further (combine like terms, evaluate if needed), (5) handle special cases (zero→1, negative→reciprocal). Common mistakes: multiplying bases when should add exponents, confusing rules (adding when should multiply for powers), working with different bases (2³ × 3² can't simplify using exponent rules).

This tests applying integer exponent properties: product rule (aᵐ × aⁿ = aᵐ⁺ⁿ), quotient rule (aᵐ ÷ aⁿ = aᵐ⁻ⁿ), power rule ((aᵐ)ⁿ = aᵐⁿ), zero exponent (a⁰=1), and negative exponents (a⁻ⁿ=1/aⁿ). For expression 3⁴ × 3⁻², apply product rule: same base 3, add exponents 4+(-2)=2, giving 3², then evaluate: 3²=9. For (2³)⁴, multiply exponents: 3×4=12 giving 2¹², evaluating: 2¹²=4096. Key principle: properties apply when bases are identical; different bases (2³ × 3²) can't combine using these rules. For $(\left(6^0$ \times $8^2$\right) \div $8^{-1}$), first evaluate $(6^0$ = 1), so 1 × $8^2$ = $8^2$, then apply quotient rule: 2 - (-1) = 3, giving $(8^3$). Choice C is correct because it correctly evaluates a⁰=1, applies the product rule implicitly, and then the quotient rule by subtracting exponents. Choice D multiplies bases instead of adding exponents (6×8=48, wrong). Strategy: (1) identify operation (multiply/divide/power of power), (2) check bases match (must be same for properties), (3) apply correct rule (add/subtract/multiply exponents), (4) simplify further (combine like terms, evaluate if needed), (5) handle special cases (zero→1, negative→reciprocal). Common mistakes: multiplying bases when should add exponents, confusing rules (adding when should multiply for powers), working with different bases (2³ × 3² can't simplify using exponent rules).

This tests applying integer exponent properties: product rule (aᵐ × aⁿ = aᵐ⁺ⁿ), quotient rule (aᵐ ÷ aⁿ = aᵐ⁻ⁿ), power rule ((aᵐ)ⁿ = aᵐⁿ), zero exponent (a⁰=1), and negative exponents (a⁻ⁿ=1/aⁿ). For expression 3⁴ × 3⁻², apply product rule: same base 3, add exponents 4+(-2)=2, giving 3², then evaluate: 3²=9. For (2³)⁴, multiply exponents: 3×4=12 giving 2¹², evaluating: 2¹²=4096. Key principle: properties apply when bases are identical; different bases (2³ × 3²) can't combine using these rules. For $(\left(9^2$$\right)^0$ \times $9^{-4}$), first apply the power rule and zero exponent: $(9^2$$)^0$ = 1, then multiply by $9^{-4}$, giving $9^{-4}$. Choice C is correct because it properly applies the power rule to get zero exponent equaling 1 and then the product rule effectively. Choice A treats the whole as $9^0$ when the zero is only on the power of power. Strategy: (1) identify operation (multiply/divide/power of power), (2) check bases match (must be same for properties), (3) apply correct rule (add/subtract/multiply exponents), (4) simplify further (combine like terms, evaluate if needed), (5) handle special cases (zero→1, negative→reciprocal). Common mistakes: multiplying bases when should add exponents, confusing rules (adding when should multiply for powers), working with different bases (2³ × 3² can't simplify using exponent rules).

This tests applying integer exponent properties: product rule ($a^m \times a^n = a^{m+n}$), quotient rule ($a^m \div a^n = a^{m-n}$), power rule (($a^m)^n = a^{m n}$), zero exponent ($a^0 = 1$), and negative exponents ($a^{-n} = \frac{1}{a^n}$). For expression $3^4 \times 3^{-2}$, apply product rule: same base 3, add exponents 4 + (-2) = 2, giving $3^2$, then evaluate: $3^2 = 9$. For ($2^3)^4$, multiply exponents: 3 × 4 = 12 giving $2^{12}$, evaluating: $2^{12} = 4096$. Key principle: properties apply when bases are identical; different bases ($2^3 \times 3^2$) can't combine using these rules. For ($\left(2^{-3}\right)^2 \div 2^{-1}$), first apply the power rule: -3 × 2 = -6, giving $2^{-6}$, then quotient rule: -6 - (-1) = -5, resulting in $2^{-5}$. Choice B is correct because it properly applies the power rule by multiplying exponents and then the quotient rule by subtracting, handling negatives correctly. Choice A adds when should multiply in power of power ($(-3) + 2 = -1$ instead of -6). Strategy: (1) identify operation (multiply/divide/power of power), (2) check bases match (must be same for properties), (3) apply correct rule (add/subtract/multiply exponents), (4) simplify further (combine like terms, evaluate if needed), (5) handle special cases (zero → 1, negative → reciprocal). Common mistakes: multiplying bases when should add exponents, confusing rules (adding when should multiply for powers), working with different bases ($2^3 \times 3^2$ can't simplify using exponent rules).

This tests applying integer exponent properties: product rule ($a^{m} \times a^{n} = a^{m+n}$), quotient rule ($a^{m} \div a^{n} = a^{m-n}$), power rule (($a^{m})^{n} = a^{m n}$), zero exponent ($a^{0}=1$), and negative exponents ($a^{-n}=1/a^{n}$). For expression $3^{4} \times 3^{-2}$, apply product rule: same base 3, add exponents 4+(-2)=2, giving $3^{2}$, then evaluate: $3^{2}=9$. For $(2^{3})^{4}$, multiply exponents: 3×4=12 giving $2^{12}$, evaluating: $2^{12}=4096$. Key principle: properties apply when bases are identical; different bases ($2^{3} \times 3^{2}$) can't combine using these rules. For ($3^{4} \times 3^{-2}$), apply the product rule: add exponents 4 + (-2) = 2, giving ($3^{2}$). Choice A is correct because it properly applies the product rule by adding exponents, including the negative one. Choice C ignores negative making $3^{4} \times 3^{-2} = 3^{-8}$ by subtracting instead of adding. Strategy: (1) identify operation (multiply/divide/power of power), (2) check bases match (must be same for properties), (3) apply correct rule (add/subtract/multiply exponents), (4) simplify further (combine like terms, evaluate if needed), (5) handle special cases (zero→1, negative→reciprocal). Common mistakes: multiplying bases when should add exponents, confusing rules (adding when should multiply for powers), working with different bases ($2^{3} \times 3^{2}$ can't simplify using exponent rules).

This tests applying integer exponent properties: product rule (aᵐ × aⁿ = aᵐ⁺ⁿ), quotient rule (aᵐ ÷ aⁿ = aᵐ⁻ⁿ), power rule ((aᵐ)ⁿ = aᵐⁿ), zero exponent (a⁰=1), and negative exponents (a⁻ⁿ=1/aⁿ). For expression 3⁴ × 3⁻², apply product rule: same base 3, add exponents 4+(-2)=2, giving 3², then evaluate: 3²=9. For (2³)⁴, multiply exponents: 3×4=12 giving 2¹², evaluating: 2¹²=4096. Key principle: properties apply when bases are identical; different bases (2³ × 3²) can't combine using these rules. For $(5^7$ \div $5^3$), apply the quotient rule: subtract exponents 7 - 3 = 4, giving $(5^4$). Choice C is correct because it properly applies the quotient rule by subtracting exponents. Choice A adds when should subtract in quotient or confuses with product rule. Strategy: (1) identify operation (multiply/divide/power of power), (2) check bases match (must be same for properties), (3) apply correct rule (add/subtract/multiply exponents), (4) simplify further (combine like terms, evaluate if needed), (5) handle special cases (zero→1, negative→reciprocal). Common mistakes: multiplying bases when should add exponents, confusing rules (adding when should multiply for powers), working with different bases (2³ × 3² can't simplify using exponent rules).

This tests applying integer exponent properties: product rule (aᵐ × aⁿ = aᵐ⁺ⁿ), quotient rule (aᵐ ÷ aⁿ = aᵐ⁻ⁿ), power rule ((aᵐ)ⁿ = aᵐⁿ), zero exponent (a⁰=1), and negative exponents (a⁻ⁿ=1/aⁿ). For expression 3⁴ × 3⁻², apply product rule: same base 3, add exponents 4+(-2)=2, giving 3², then evaluate: 3²=9. For (2³)⁴, multiply exponents: 3×4=12 giving 2¹², evaluating: 2¹²=4096. Key principle: properties apply when bases are identical; different bases (2³ × 3²) can't combine using these rules. For $(7^0$ \times $2^5$), first recognize the zero exponent gives 1, then compute $(2^5$ = 32), so 1 × 32 = 32. Choice C is correct because it correctly evaluates a⁰=1 and then multiplies by the evaluated power. Choice A treats a⁰ as 0 when equals 1. Strategy: (1) identify operation (multiply/divide/power of power), (2) check bases match (must be same for properties), (3) apply correct rule (add/subtract/multiply exponents), (4) simplify further (combine like terms, evaluate if needed), (5) handle special cases (zero→1, negative→reciprocal). Common mistakes: multiplying bases when should add exponents, confusing rules (adding when should multiply for powers), working with different bases (2³ × 3² can't simplify using exponent rules).

This tests applying integer exponent properties: product rule ($a^m \times a^n = a^{m+n}$), quotient rule ($a^m \div a^n = a^{m-n}$), power rule ($(a^m)^n = a^{m n}$), zero exponent ($a^0 = 1$), and negative exponents ($a^{-n} = \frac{1}{a^n}$). For expression $3^4 \times 3^{-2}$, apply product rule: same base 3, add exponents 4 + (-2) = 2, giving $3^2$, then evaluate: $3^2 = 9$. For $(2^3)^4$, multiply exponents: 3 \times 4 = 12 giving $2^{12}$, evaluating: $2^{12} = 4096$. Key principle: properties apply when bases are identical; different bases ($2^3 \times 3^2$) can't combine using these rules. For $(\left(4^{-2}\right)^3)$, apply the power rule: multiply exponents -2 \times 3 = -6, giving $4^{-6}$. Choice A is correct because it properly applies the power rule by multiplying exponents, including the negative one. Choice D ignores negative making $4^{-6} = 4^6$ when should be $1/4^6$. Strategy: (1) identify operation (multiply/divide/power of power), (2) check bases match (must be same for properties), (3) apply correct rule (add/subtract/multiply exponents), (4) simplify further (combine like terms, evaluate if needed), (5) handle special cases (zero→1, negative→reciprocal). Common mistakes: multiplying bases when should add exponents, confusing rules (adding when should multiply for powers), working with different bases ($2^3 \times 3^2$ can't simplify using exponent rules).