To simplify the expression \( 150 \cdot 7^{-2} \cdot (-4)^{-3} \) while converting all exponents to positive, we can rewrite it as follows:
- The term \( 7^{-2} \) can be expressed as \( \frac{1}{7^2} \).
- 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:
- \( \frac{(-4)^3}{7^2} \): This does not represent the original expression because it has the negative base in the numerator.
- \( \frac{7^2}{(-4)^3} \): This is incorrect as it swaps the numerator and denominator.
- \( \frac{4^3}{7^2} \): This is incorrect as it changes the base from \((-4)^3\) to \(4^3\), which changes the value of the expression.
- \( \frac{1}{72 \cdot (-4)^{-3}} \): 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.