Which of the following is an equivalent expression to 150 ⋅ 7−2 (−4)−3 with only positive exponents that has been generated by applying the Properties of Integer Exponents?(1 point)
Responses
72(−4)3
172 ⋅ (−4)−3
4372
(−4)372
The correct answer is 4372. This is obtained by applying the product of powers rule, which states that (a^b) * (a^c) = a^(b+c). In the expression 150 ⋅ 7^(-2) * (-4)^(-3), we can change the negative exponents to positive exponents by moving the base to the denominator with a positive exponent.
150 ⋅ 7^(-2) * (-4)^(-3) = 150 * (1/7^2) * (1/(-4)^3)
Now, we can apply the product of powers rule to combine the terms:
150 * (1/7^2) * (1/(-4)^3) = 150 * (1/(49)) * (1/(-64))
Simplifying further, we get:
150 * (1/49) * (1/(-64)) = 150/(-3136)
This can be written as a positive exponent by changing the fraction to a negative exponent:
150/(-3136) = 150 * (-3136)^(-1)
Finally, this expression can be simplified to:
150 * (-3136)^(-1) = 4372
To find the equivalent expression with only positive exponents using the Properties of Integer Exponents, we can apply the rule:
a^(-n) = 1/(a^n)
Let's break down the given expression step-by-step:
150 ⋅ 7^(-2) • (-4)^(-3)
First, let's apply the rule to both exponents:
150 ⋅ 1/(7^2) • 1/((-4)^3)
Next, simplify the exponents:
150 ⋅ 1/49 • 1/(-64)
Now, simplify the expression:
150/(49 * (-64))
= 150/(-3136)
The equivalent expression with only positive exponents is:
1/(3136/150)
= 150/3136
Therefore, the correct option is:
4372 (option C)