Question

an equivalent expression to 5^9⋅5^−13 with only positive exponents that has been generated by applying the Properties of Integer Exponents?

Answer 1

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

the Property of Negative Exponents and the Product Rule of Exponents
the Property of Negative Exponents and the Product Rule of Exponents

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

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

only the Product Rule of Exponents

Answer 2

To simplify the expression \( 5^9 \cdot 5^{-13} \) using the Properties of Integer Exponents, we can apply the property that states when you multiply two expressions with the same base, you add the exponents:

\[
5^9 \cdot 5^{-13} = 5^{9 + (-13)} = 5^{9 - 13} = 5^{-4}
\]

Now, we want to express this with only positive exponents. To convert \( 5^{-4} \) to an expression with a positive exponent, we use the rule that states \( a^{-n} = \frac{1}{a^n} \):

\[
5^{-4} = \frac{1}{5^4}
\]

Thus, the equivalent expression to \( 5^9 \cdot 5^{-13} \) with only positive exponents is:

\[
\frac{1}{5^4}
\]

Answer 3

To generate the equivalent expression \( 3^{14} \) from \( \frac{3^5}{3^{-9}} \), we can use the following properties of exponents:

1. **Property of Negative Exponents**: This property states that \( a^{-n} = \frac{1}{a^n} \). Therefore, \( \frac{1}{3^{-9}} \) can be rewritten as \( 3^{9} \).

2. **Quotient Rule of Exponents**: This property states that when you divide two expressions with the same base, you subtract the exponents: \( \frac{a^m}{a^n} = a^{m - n} \).

Using these properties, we can simplify the expression:

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

Thus, both the Property of Negative Exponents and the Product Rule (which is often a simpler term for the Quotient Rule) are involved in this transformation.

Therefore, the correct response would be:

**the Property of Negative Exponents and the Product Rule of Exponents**.