Final answer:
Jill made a mistake by incorrectly multiplying bases while dealing with exponents. The correction involves recognizing that 4 is 2 squared and then applying the rules of exponents correctly to get a final answer of 2^9, not 8^6.
Step-by-step explanation:
Jill writes 2^3 • 4^3 = 8^6, and the teacher marked it wrong. Let's analyze where the mistake lies. The error in Jill's statement is that she incorrectly multiplied the bases. When simplifying expressions with exponents, the properties of exponents dictate different operations depending on the situation. In the case of multiplying powers with the same base, we keep the base and add the exponents. However, when the bases are different, we cannot combine them into a single power by simply adding or multiplying the exponents. In Jill's case, she should have rewritten 4^3 as (2^2)^3 to have the same base and applied the rules accordingly.
Correctly solving the expression involves recognizing that 4 is a power of 2, specifically 2^2, therefore 4^3 is (2^2)^3. According to the rules of exponents, when raising a power to a power, you multiply the exponents, giving us 2^(2×3) which simplifies to 2^6. Thus, when we multiply that by 2^3, we should add the exponents according to the rule for multiplying exponential terms with the same base, resulting in a final answer of 2^9. Hence, the correct way to multiply 2^3 and 4^3 is 2^3 × (2^2)^3 which equals 2^3 × 2^6, and not 8^6.