Apply the Properties of Integer Exponents to generate equivalent expressions to
(4^3 ⋅ 7^−2)^−5/7^9 ⋅ 4^−13 with only positive exponents, then solve the expression. Your answer will be a simplified fraction with no remaining exponents.
To generate equivalent expressions with only positive exponents, we can apply the following properties of integer exponents:
1. (a^m * b^n)^p = a^(m*p) * b^(n*p)
2. a^(-m) = 1/a^m
Applying these properties, we have:
(4^3 * 7^(-2))^(-5) / (7^9 * 4^(-13))
= (4^(3 * -5) * 7^(-2 * -5)) / (7^9 * 4^(-13))
= 4^(-15) * 7^10 / (7^9 * 4^(-13))
Now, let's simplify the expression further by using the property a^m / a^n = a^(m-n):
= 4^(-15 - (-13)) * 7^(10 - 9)
= 4^(-15 + 13) * 7^1
= 4^(-2) * 7
Using the property a^(-m) = 1/a^m, we simplify further:
= (1/4^2) * 7
= 1/16 * 7
= 7/16
Therefore, the simplified fraction with no remaining exponents is 7/16.
Which of the following is an equivalent expression to 5^9⋅5^−13 with only positive exponents that has been generated by applying the Properties of Integer Exponents?
Responses
5^9/5^−13
Start Fraction 5 superscript 9 baseline over 5 superscript negative 13 baseline End Fraction
1/625
Start Fraction 1 over 625 end fraction
5^4
5 superscript 4 baseline
1/5^4
To find an equivalent expression with only positive exponents, we can apply the property of integer exponents that states a^(-m) = 1/a^m.
Applying this property to 5^(-13), we get 1/5^13. Therefore, the equivalent expression with only positive exponents is:
5^9 * 1/5^13
To simplify this expression, we can use the property a^m / a^n = a^(m-n). In this case, m = 9 and n = 13.
Therefore, our final simplified expression is:
5^(9 - 13) = 5^(-4)
So, the correct equivalent expression is 1/5^4.
Which property of exponents was used to generate the equivalent expression 3^14 from 3^5/3^−9?
Responses
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 Property of Negative Exponents and the Product Rule of Exponents
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.
Which property of exponents was used to generate the equivalent expression 3^14 from 3^5/3^−9?
Responses
A only the Power Rule of Exponents
B only the Product Rule of Exponents
C the Property of Negative Exponents and the Product Rule of Exponents
D only the Property of Negative Exponents