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Original Question
Which property of exponents was used to generate the equivalent expression 3^14 from 3^5/3^−9?(1 point) Responses only the Powe...Asked by Ame
Which property of exponents was used to generate the equivalent expression 314 from 353−9 ?(1 point)
Responses
only the Product Rule of Exponents
only the Power Rule of Exponents
only the Property of Negative Exponents
the Property of Negative Exponents and the Product Rule of Exponents
Responses
only the Product Rule of Exponents
only the Power Rule of Exponents
only the Property of Negative Exponents
the Property of Negative Exponents and the Product Rule of Exponents
Answers
Answered by
GPT-4o mini
To simplify the expression \( 35^{3} \cdot 35^{-9} \) to \( 35^{14} \), we can use the **Product Rule of Exponents**, which states that when multiplying two expressions with the same base, you add the exponents:
\[
a^m \cdot a^n = a^{m+n}
\]
In this case:
\[
35^{3} \cdot 35^{-9} = 35^{3 + (-9)} = 35^{-6}
\]
If you wanted to express \( 35^{-6} \) as a positive exponent, you'd apply the **Property of Negative Exponents**, which states that:
\[
a^{-m} = \frac{1}{a^{m}}
\]
Thus, \( 35^{-6} = \frac{1}{35^{6}} \).
So, the properties applied to generate the equivalent expression \( 35^{-6} \) from \( 35^{3} \cdot 35^{-9} \) are the **Property of Negative Exponents** and the **Product Rule of Exponents**.
Therefore, the correct response is:
**the Property of Negative Exponents and the Product Rule of Exponents**.
\[
a^m \cdot a^n = a^{m+n}
\]
In this case:
\[
35^{3} \cdot 35^{-9} = 35^{3 + (-9)} = 35^{-6}
\]
If you wanted to express \( 35^{-6} \) as a positive exponent, you'd apply the **Property of Negative Exponents**, which states that:
\[
a^{-m} = \frac{1}{a^{m}}
\]
Thus, \( 35^{-6} = \frac{1}{35^{6}} \).
So, the properties applied to generate the equivalent expression \( 35^{-6} \) from \( 35^{3} \cdot 35^{-9} \) are the **Property of Negative Exponents** and the **Product Rule of Exponents**.
Therefore, the correct response is:
**the Property of Negative Exponents and the Product Rule of Exponents**.
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