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 Power Rule of Exponents
only the Power Rule of Exponents
only the Property of Negative Exponents
only the Property of Negative Exponents
only the Product Rule of Exponents
only the Property of Negative Exponents
Which of the following is an equivalent expression to 15^0 * 7^−2/(−4)^−3 with only positive exponents that has been generated by applying the Properties of Integer Exponents? (1 point)
Responses
4^3/7^2
Start Fraction 4 cubed over 7 squared end fraction
1/7^2 * (−4)^−3
Start Fraction 1 over 7 squared times left parenthesis negative 4 right parenthesis superscript negative 3 baseline End Fraction
7^2/(−4)^3
Start Fraction 7 squared over left parenthesis negative 4 right parenthesis cubed end fraction
(−4)^3/7^2
start fraction left parenthesis negative 4 right parenthesis cubed over 7 squared end fraction
(−4)^3/7^2
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.
To generate the equivalent expression 3^14 from 3^5/3^−9, we need to understand the properties of exponents used in the process.
The Property of Negative Exponents states that for any nonzero number a, a^(-n) = 1/a^n. It allows us to convert a negative exponent into a positive exponent by taking the reciprocal of the base.
The Product Rule of Exponents states that for any nonzero number a, a^n * a^m = a^(n + m). It allows us to multiply two exponents with the same base by adding their powers.
The Power Rule of Exponents states that for any nonzero number a, (a^n)^m = a^(n * m). It allows us to raise an exponent to another exponent by multiplying their powers.
When looking at the given expression 3^5/3^-9, we can simplify it step by step using the properties of exponents:
1. Apply the Property of Negative Exponents to the denominator:
3^5/3^-9 = 3^5 * (1/3^9)
2. Simplify the fraction:
3^5 * (1/3^9) = 3^5 * 3^(-9)
3. Apply the Product Rule of Exponents:
3^5 * 3^(-9) = 3^(5 + (-9))
4. Simplify the exponent:
3^(5 + (-9)) = 3^(-4)
Therefore, we can see that only the Property of Negative Exponents was used to generate the equivalent expression 3^14 from 3^5/3^-9.