Which property of exponents was used to generate the equivalent expression 3 to the power of 14 from 3 to the power of 5 over 3 to the power of −9 ?(1 point)

Question

Responses

only the Power Rule of Exponents
only the Power Rule of Exponents

only the Product Rule of Exponents
only the Product Rule of Exponents

only the Property of Negative Exponents
only the Property of Negative Exponents

the Property of Negative Exponents and the Product Rule of Exponents

Answer 1

To generate the expression \(3^{14}\) from \( \frac{3^5}{3^{-9}} \), we can use the Quotient Rule of Exponents, which states that when dividing two expressions with the same base, we subtract the exponents.

Applying this rule:

\[ \frac{3^5}{3^{-9}} = 3^{5 - (-9)} = 3^{5 + 9} = 3^{14} \]

So the correct response is not listed, but the closest correct reasoning involves understanding that the Quotient Rule of Exponents is in effect since we are dividing one exponent by another, also involving recognizing the negative exponent which indicates the need to switch the base to the reciprocal.

If we are isolating just the given options without including the Quotient Rule:

• Since the change from \(3^{-9}\) to its positive equivalent involves understanding the Property of Negative Exponents, that aspect is considered.

Thus, the closest acceptable response from the options provided is the Property of Negative Exponents. However, it's important to point out that technically, the operation performed was the application of the Quotient Rule of Exponents directly.