To simplify the expression \( 150 \cdot 7^{-2} \cdot (-4)^{-3} \) with only positive exponents, we can apply the Properties of Integer Exponents.
Firstly, let's carry out the simplification step-by-step:
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Rewrite using positive exponents:
- \( 7^{-2} = \frac{1}{7^2} \)
- \( (-4)^{-3} = \frac{1}{(-4)^3} \)
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Substituting back: \[ 150 \cdot \frac{1}{7^2} \cdot \frac{1}{(-4)^3} = \frac{150}{7^2 \cdot (-4)^3} \]
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Calculating \( (-4)^3 \): \[ (-4)^3 = -64 \quad \text{(so we will have $\frac{150}{7^2 \cdot (-64)}$)} \]
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Putting it all together, we have: \[ \frac{150}{7^2 \cdot -64} \]
Now let's evaluate each option given:
- 4372 - This is just a number, not in a fraction form.
- Start Fraction 4 cubed over 7 squared end fraction - This is \(\frac{4^3}{7^2}\), not equivalent to our original expression.
- 72(−4)3 - This isn't correctly structured and simplifies not to the same form.
- Start Fraction 7 squared over left parenthesis negative 4 right parenthesis cubed end fraction - This is \(\frac{7^2}{(-4)^3}\), corresponding to the form of our expression.
- 172 ⋅ (−4)−3 - This is not simplified to positive exponents.
The correct equivalent expression is: \[ \frac{150}{7^2 \cdot (-4)^3} = \frac{150}{7^2} \cdot \frac{1}{(-64)} \] Which correlates to the representation with fractions.
So the equivalent expression with only positive exponents in simplified form aligns most closely with this option given: Start Fraction 7 squared over left parenthesis negative 4 right parenthesis cubed end fraction.