Which property of exponents was used to generate the equivalent expression 3^14 from 3^5/3^−9?(1 point)
Responses
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
only the Product Rule of Exponents
the Property of Negative Exponents and the Product Rule of Exponents
The correct answer is: only the Property of Negative Exponents.
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).
The property of exponents used to generate the equivalent expression 3^14 from 3^5/3^−9 is the Property of Negative Exponents and the Product Rule of Exponents.
To understand this, let's break it down step by step:
1. Property of Negative Exponents: According to this property, any non-zero number raised to a negative exponent can be rewritten as its reciprocal raised to the positive exponent. For example, a^(-n) = 1/a^n.
In the given expression, we have 3^(-9) in the denominator. To apply the Property of Negative Exponents, we can rewrite 3^(-9) as 1/3^9.
Now the expression becomes 3^5 / (1/3^9).
2. Product Rule of Exponents: According to this rule, when dividing two numbers with the same base raised to different exponents, you can subtract the exponents. In other words, a^n / a^m = a^(n-m).
Now, using the Product Rule of Exponents, we can simplify the expression further:
3^5 / (1/3^9) = 3^5 * 3^9.
3. By multiplying the two expressions with the same base, we can add the exponents:
3^5 * 3^9 = 3^(5+9) = 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.