1 answer
(click or scroll down)
Bot GPT 4-o mini To simplify the expression \( 150 \cdot 7^{-2} \cdot (-4)^{-3} \) and express it with only positive exponents, we can use the properties of integer exponents.
1. Rewrite the negative exponents as follows:
- \( 7^{-2} \) becomes \( \frac{1}{7^{2}} \)
- \( (-4)^{-3} \) becomes \( \frac{1}{(-4)^{3}} \)
Thus, we can rewrite the expression:
\[
150 \cdot \frac{1}{7^{2}} \cdot \frac{1}{(-4)^{3}} = \frac{150}{7^{2} \cdot (-4)^{3}}
\]
Now we have expressed everything with positive exponents.
Next, we can combine the terms in the denominator:
\[
\frac{150}{7^{2} \cdot (-4)^{3}} = \frac{150}{7^{2} \cdot (-64)}
\]
However, to match your options with only positive exponents correctly, we will mainly focus on how the options are structured.
Let’s look at the given outcomes:
1. \( 72(-4)^{3} \)
2. \( \frac{7^{2}}{(-4)^{3}} \)
3. \( (-4)^{3} \frac{7^{2}}{} \)
4. \( 172 \cdot (-4)^{-3} \)
5. \( \frac{4}{7^{2}} \)
Given the format of your expressions, we see the common format would be the positive forms.
Thus, our equivalent expression in positive exponents is:
\[
\frac{150}{7^{2} \cdot (-4)^{3}} \quad \text{which is equivalent to} \quad \frac{150}{7^{2}} \cdot \frac{1}{(-4)^{3}}
\]
Since none of those seem directly listed, we'd have to look closely at what you gave and conclude:
- \( \boxed{\frac{1}{7^{2} \cdot (-4)^{3}}} \) or simply \( 72(-4)^{3} \) — though, this particular expression with \( 150 \) isn't fulfilled correctly. I would re-check if any are mathematically valid here considering their formation against standard values.
Make sure to cross-examine each of your options if they match this standardized structure with focus strictly on positive exponents!