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

Responses

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

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

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

4^3/7^2

Answer 1

To simplify the expression \( \frac{150 \cdot 7^{-2}}{(-4)^{-3}} \), we will apply the properties of integer exponents stepwise.

1. Rewrite the expression while moving the negative exponents to the denominator: \[ \frac{150 \cdot 7^{-2}}{(-4)^{-3}} = 150 \cdot \frac{1}{7^2} \cdot (-4)^3 = \frac{150 \cdot (-4)^3}{7^2} \]

2. Now, calculating \( (-4)^3 \): \[ (-4)^3 = -64 \]

3. So, we replace \( (-4)^3 \) in the expression: \[ = \frac{150 \cdot (-64)}{7^2} \]

4. Converting this to a fraction, we can simplify it: \[ = \frac{-9600}{49} \]

5. At this point, all the exponents are positive since we have \( 7^2 \) in the denominator and a positive base for \( 4 \).

Now, checking the choices provided:

1. 172 ⋅ (−4)−3 - This has a negative exponent.
2. Start Fraction 1 over 7 squared times left parenthesis negative 4 right parenthesis superscript negative 3 baseline End Fraction - This has negative exponents.
3. 72(−4)3 - This has no negative exponents but does not reflect the original expression's equivalence correctly.
4. Start Fraction 7 squared over left parenthesis negative 4 right parenthesis cubed end fraction - This has negative in the denominator.
5. (−4)372 - This has negative exponents and doesn't capture the conversion correctly.
6. 4^3/7^2 - This presents only positive exponents.

Therefore, the equivalent expression to \( \frac{150 \cdot 7^{-2}}{(-4)^{-3}} \) with only positive exponents is:

4^3/7^2