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)

(−4)^3/7^2

7^2/(−4)^3

4^3/7^2

1/72 ⋅ (-4)^-3

Answer 1

To simplify the expression \( 150 \cdot 7^{-2} \cdot (-4)^{-3} \) while converting all exponents to positive, we can rewrite it as follows:

1. The term \( 7^{-2} \) can be expressed as \( \frac{1}{7^2} \).
2. The term \( (-4)^{-3} \) can be expressed as \( \frac{1}{(-4)^3} \).

Now we can substitute these back into the expression:

\[
150 \cdot \frac{1}{7^2} \cdot \frac{1}{(-4)^3} = \frac{150}{7^2 \cdot (-4)^3}
\]

Any expression that represents this with only positive exponents would need to be equivalent.

Now, let's review the options provided:

1. **\( \frac{(-4)^3}{7^2} \)**: This does not represent the original expression because it has the negative base in the numerator.
2. **\( \frac{7^2}{(-4)^3} \)**: This is incorrect as it swaps the numerator and denominator.
3. **\( \frac{4^3}{7^2} \)**: This is incorrect as it changes the base from \((-4)^3\) to \(4^3\), which changes the value of the expression.
4. **\( \frac{1}{72 \cdot (-4)^{-3}} \)**: This one has \( (-4)^{-3} \) in the denominator, which still has a negative exponent.

As we have already calculated that the simplified form is

\[
\frac{150}{7^2 \cdot (-4)^3}
\]

None of the provided options match or represent this equivalent expression correctly.

However, if we evaluate the expression directly, we can substitute \(150\) as \(72 \cdot \frac{25}{72}\), making **none** of the given options equivalent using only positive exponents.

Based on what's provided, there is no option that fits the criteria for equivalent expression with only positive exponents.