Solve this---**Question Stem:** Which number is equivalent to **Mathematical Expression:** 3^4 / 3^2 **Options:** A 2 B 9 C 81 D 729

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To solve this problem, we need to simplify the expression 3 to the power of 4 divided by 3 to the power of 2. When dividing exponential expressions with the same base, we subtract the exponents. Using the rule a to the power of m divided by a to the power of n equals a to the power of m minus n. In our case, we have 3 to the power of 4 minus 2, which equals 3 squared. Calculating 3 squared gives us 3 times 3, which equals 9. Looking at our options, the answer is B, 9. Let's solve this step by step. First, we identify our expression: 3 to the power of 4 divided by 3 to the power of 2. Next, we apply the division rule for exponents: when dividing powers with the same base, we subtract the exponents. So, 3 to the power of 4 divided by 3 to the power of 2 equals 3 to the power of 4 minus 2, which is 3 squared. Now we calculate 3 squared, which is 3 times 3, giving us 9. Looking at the options, we can see that 9 corresponds to option B. Therefore, 3 to the power of 4 divided by 3 to the power of 2 equals 9. Let's compare all the options to find the correct answer. For Option A, which is 2, we calculate 3 to the power of 4 divided by 3 to the power of 2, which equals 3 squared or 9. Since 9 is not equal to 2, Option A is incorrect. For Option B, which is 9, we get 3 squared equals 9, which matches. So Option B is correct. For Option C, which is 81, we know that 3 squared equals 9, not 81. So Option C is incorrect. For Option D, which is 729, we again have 3 squared equals 9, not 729. So Option D is also incorrect. Looking at our table of powers of 3, we can see that 3 to the power of 2 equals 9, 3 to the power of 4 equals 81, and 3 to the power of 6 equals 729. Therefore, the correct answer is Option B, 9. Let's review the key exponent rules that are essential for solving problems like this. The multiplication rule states that when multiplying powers with the same base, we add the exponents. The division rule, which we used in our problem, states that when dividing powers with the same base, we subtract the exponents. The power rule tells us that when raising a power to another power, we multiply the exponents. The product rule states that when raising a product to a power, we raise each factor to that power. The quotient rule is similar for fractions. The zero exponent rule states that any number raised to the power of zero equals 1. And the negative exponent rule tells us that a negative exponent means we take the reciprocal. Looking at some examples: For 2 cubed times 2 squared, we add the exponents to get 2 to the power of 5, which equals 32. For 5 to the power of 4 divided by 5 squared, we subtract the exponents to get 5 squared, which equals 25. For 2 cubed, all squared, we multiply the exponents to get 2 to the power of 6, which equals 64. And for our problem, 3 to the power of 4 divided by 3 squared, we subtract the exponents to get 3 squared, which equals 9. Let's summarize what we've learned. When dividing powers with the same base, we subtract the exponents. This is expressed by the rule: a to the power of m divided by a to the power of n equals a to the power of m minus n. For our specific problem, 3 to the power of 4 divided by 3 to the power of 2, we apply this division rule to get 3 to the power of 4 minus 2, which equals 3 squared, which equals 9. Therefore, the correct answer is option B, 9. Understanding exponent rules is essential for simplifying algebraic expressions and solving more complex mathematical problems. Always verify your answer by checking all options and applying the correct rule. This methodical approach ensures accuracy in your calculations.