Question

Which property of exponents was used to generate the equivalent expression of three with the power of 14 from three with the power of five over three with a negative power of nine with the answer I’ll need the product role of exponents or the negative exponents and the product role of exponents or only the power role of exponents or would it be only the property of negative exponents

Answer 1

To analyze the expression you provided, let's break it down step by step. You have the expression \( \frac{3^5}{3^{-9}} \).

To simplify this expression, you can use the property of exponents which states that when you divide two expressions with the same base, you subtract the exponents:

\[ \frac{a^m}{a^n} = a^{m-n} \]

In your case, this means:

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

From this, we can conclude that the property used to generate the equivalent expression of \( 3^{14} \) from \( \frac{3^5}{3^{-9}} \) is the quotient rule of exponents (which involves subtraction of exponents) combined with the handling of the negative exponent.

So, the correct answer is the quotient rule of exponents (or the property of exponents). The negative exponent property is also relevant here because \( 3^{-9} \) can be rewritten as \( \frac{1}{3^9} \), but the primary rule applied for this simplification is the quotient rule of exponents.