The Correct Answer and Explanation is:
The correct answer is 3³.
This problem asks to simplify the expression 3⁻⁹ / 3⁻¹². To solve this, we use the properties of exponents, specifically the quotient rule for powers with the same base.
The quotient rule of exponents states that when dividing two terms with the same base, you subtract the exponent of the denominator from the exponent of the numerator. The general formula is aᵐ / aⁿ = a⁽ᵐ⁻ⁿ⁾.
In the given expression, the base ‘a’ is 3. The exponent in the numerator ‘m’ is -9, and the exponent in the denominator ‘n’ is -12. Applying the rule, we set up the subtraction in the exponent:
3⁽⁻⁹ ⁻ ⁽⁻¹²⁾⁾
It is crucial to correctly handle the subtraction of a negative number. Subtracting a negative is the same as adding a positive. Therefore, the calculation for the new exponent becomes:
-9 + 12 = 3
This means the entire expression simplifies to 3³.
An alternative method involves first converting the negative exponents into positive ones using the rule a⁻ⁿ = 1/aⁿ. The expression 3⁻⁹ / 3⁻¹² can be rewritten as (1/3⁹) / (1/3¹²). To divide by a fraction, we multiply by its reciprocal. This gives us (1/3⁹) × (3¹²/1), which is equal to 3¹²/3⁹. Applying the quotient rule to this form gives 3⁽¹²⁻⁹⁾, which also equals 3³.
Both methods confirm that the simplified form of the expression is 3³. Looking at the provided multiple-choice options, 3³ is the third choice.
