To find an equivalent expression to \( 150 \cdot 7^{-2} \cdot (-4)^{-3} \) with only positive exponents, we can apply the properties of integer exponents.
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First, we rewrite the expression:
\[ 150 \cdot 7^{-2} \cdot (-4)^{-3} \]
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The term \( 7^{-2} \) can be rewritten using the property \( a^{-n} = \frac{1}{a^n} \):
\[ 150 \cdot \frac{1}{7^2} \cdot (-4)^{-3} \]
Which simplifies to:
\[ \frac{150}{7^2} \cdot (-4)^{-3} \]
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The term \( (-4)^{-3} \) can also be rewritten as:
\[ (-4)^{-3} = \frac{1}{(-4)^3} \]
Substituting back, we get:
\[ \frac{150}{7^2} \cdot \frac{1}{(-4)^3} \]
Which simplifies to:
\[ \frac{150}{7^2 \cdot (-4)^3} \]
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We can further simplify the expression if needed; however, the goal is to translate it into an equivalent positive exponent format.
Now let's compare the results with the given options:
- \( 4372 \) (This doesn’t match our expression.)
- \( \frac{4^3}{7^2} \) (This does not match as well.)
- \( (-4)^3 \) (This is not the format we need to be in either.)
- \( \frac{1}{7^2 \cdot (-4)^3} \) (This option reflects our calculation accurately and has only positive exponents.)
Thus, the correct equivalent expression is:
\[ \frac{1}{7^2 \cdot (-4)^3} \]
This corresponds to option 4:
\[ \text{Start Fraction 1 over 7 squared times left parenthesis negative 4 right parenthesis superscript negative 3 baseline End Fraction} \]
3 answers