which property of exponents was used to generate the equivalent expression 3^14 from 3^5/3^-9?
a. only the Product Rule of Exponents
b. only the Property of Negative Exponents
c. only the Power Rule of Exponents
d. the Property of Negative Exponents and the Product Rule of Exponents
d. the Property of Negative Exponents and the Product Rule of Exponents
The correct answer is d. The Property of Negative Exponents and the Product Rule of Exponents were used to generate the equivalent expression 3^14 from 3^5/3^-9.
To understand how these properties were used, let's break it down step by step:
1. Start with the expression 3^5/3^-9.
2. Apply the Property of Negative Exponents, which states that a negative exponent in the denominator can be converted to a positive exponent in the numerator by flipping the base. In this case, we can move the 3^-9 from the denominator to the numerator by flipping it to 3^9.
This gives us the expression (3^5)(3^9).
3. Apply the Product Rule of Exponents, which states that when multiplying two exponential expressions with the same base, you can add their exponents together. In this case, we have 3^5 multiplied by 3^9, so we can add the exponents to get 3^(5+9).
Simplifying further, 5+9 equals 14, so the expression becomes 3^14.
Therefore, the Property of Negative Exponents and the Product Rule of Exponents were both used to generate the equivalent expression 3^14 from 3^5/3^-9.