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 this expression, $(3^{4}$ × $3^{-2}$) = $3^{4-2}$ = $3^{2}$, then ÷ $3^{3}$ = $3^{2-3}$ = $3^{-1}$. Choice B is correct because it properly applies the product rule to add exponents and the quotient rule to subtract, resulting in $3^{-1}$. Choice A is wrong because it multiplies exponents incorrectly, perhaps treating as $(3^{4-2}$$)/3^{3}$$=3^{2}$$/3^{3}$$=3^{6}$ or other confusion. 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 (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 ($a^m \times a^n = a^{m+n}$), 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: $3 \times 4 = 12$ 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 this expression, $(2^{3})^{4} = 2^{12}$, then $\div 2^{10} = 2^{12-10} = 2^{2}$. Choice A is correct because it applies the power rule by multiplying exponents and the quotient rule by subtracting, giving $2^{2}$. Choice B is wrong because it adds exponents instead of multiplying in the power rule, like $3+4=7$ then $2^{7}/2^{10}=2^{-3}$ or other error, but here it's 22 perhaps from $3 \times 4 + 10$. 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 this expression, first $(7^{2}$$)^{3}$ = $7^{6}$, then ÷ $7^{5}$ = $7^{6-5}$ = $7^{1}$. Choice A is correct because it correctly applies the power rule by multiplying exponents and the quotient rule by subtracting, giving $7^{1}$. Choice B is wrong because it adds exponents instead of multiplying in the power rule, perhaps doing 2+3=5 then dividing by $7^{5}$ for $7^{0}$ or other error. 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 ($4^{-3}$), 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 (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 $ (8^2)^3 / 8^5 $, first apply power rule: $ (8^2)^3 = 8^{2 \times 3} = 8^6 $. Then apply quotient rule: $ 8^6 \div 8^5 = 8^{6-5} = 8^1 $. Choice B ($8^1$) is correct because it properly applies the power rule (multiplying 2 \times 3 = 6) then the quotient rule (subtracting 6 - 5 = 1). Choice A ($8^{11}$) incorrectly adds 6 + 5 instead of subtracting, choice C ($8^6$) ignores the division by 8^5, and choice D ($8^{-1}$) would result from 8^5 \div 8^6. 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\to1, negative\toreciprocal). 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 $(\left(4^{-2}\right)^3)$, 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).

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).