Final answer:
Luciano's calculation of the ratio of 9^2 to 9^7 as 1/9^5 is incorrect. The proper calculation should use the rules of exponents, resulting in 9^-5, which is equivalent to 1/9^5, meaning the ratio is the reciprocal of 9^5.
Step-by-step explanation:
Luciano is not correct in his calculation of the ratio of 92 to 97. To compare the ratio of two powers with the same base, we can use the properties of exponents to simplify the expression. The ratio 92 to 97 is calculated as follows:
Therefore, the ratio of 92 to 97 is 9-5, not 1/95. The negative exponent indicates a reciprocal, so we can rewrite 9-5 as 1/95, which shows that Luciano's answer was missing the reciprocal aspect of the negative exponent.
In comparing ratios of powers with the same base, the rules of exponents dictate that xa / xb = xa-b. From this, we conclude that dividing powers with the same base requires subtracting the exponents. When the resulting exponent is negative, the reciprocal of the base raised to the positive exponent is the correct representation.