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)

Question

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

172 ⋅ (−4)−3
Start Fraction 1 over 7 squared times left parenthesis negative 4 right parenthesis superscript negative 3 baseline End Fraction

(−4)372
start fraction left parenthesis negative 4 right parenthesis cubed over 7 squared end fraction

72(−4)3
Start Fraction 7 squared over left parenthesis negative 4 right parenthesis cubed end fraction

4372

Answer 1

The equivalent expression with only positive exponents is:

(7^2)/((-4)^3)

Answer 2

To convert the expression with negative exponents to an equivalent expression with positive exponents, we can make use of the properties of integer exponents.

The property states that a negative exponent can be rewritten as the reciprocal of the base raised to the positive exponent.

Given 150 ⋅ 7^(-2) * (-4)^(-3), we can rewrite it as:
= (1/150) * (1/7^2) * (1/(-4)^3)

Now, we can simplify the expression further by evaluating the exponents:
= (1/150) * (1/49) * (1/(-64))

Further simplifying, we get:
= (1/150) * (1/49) * (-1/64)

Combining the numerators, we have:
= -1/(150 * 49 * 64)

Therefore, the equivalent expression with only positive exponents is -1/(150 * 49 * 64).

Answer 3

To find the equivalent expression with only positive exponents that has been generated by applying the Properties of Integer Exponents, we can use the general rule for exponents. According to the rule, for any nonzero number a and positive integers m and n:

(a^m)^n = a^(m*n)

Let's apply this rule to each option and simplify them:

Option 1: 172 ⋅ (−4)^(-3)
To make the exponent positive, we can apply the rule:
(−4)^(-3) = 1/((-4)^3) = 1/(-64) = -1/64

So, the expression becomes: 172 ⋅ (-1/64) = -172/64

Option 2: (−4)^3/7^2
The exponent is already positive, so no changes are required.
Therefore, the expression remains as: (-4)^3/7^2 = -64/49

Option 3: 7^2/(-4)^3
The exponent is already positive, so no changes are required.
Therefore, the expression remains as: 7^2/(-4)^3 = 49/-64

Option 4: 4^3/7^2
The exponent is already positive, so no changes are required.
Therefore, the expression remains as: 4^3/7^2 = 64/49

From the options provided, the expression with only positive exponents that has been generated by applying the Properties of Integer Exponents is:

Option 4: 64/49