To simplify the expression \( (4g^{-3})(2g^{4}) \), we can use the properties of exponents and multiplication:
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Multiply the coefficients: \[ 4 \cdot 2 = 8 \]
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Combine the terms with exponents: \[ g^{-3} \cdot g^{4} = g^{-3 + 4} = g^{1} \]
Putting it all together: \[ (4g^{-3})(2g^{4}) = 8g^{1} = 8g \]
Now let's analyze the given options:
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8g
- Equivalent to \( 8g^{1} \).
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4×g×g×g×2×g×g×g×g
- This expression expands \( 4g^{-3} \) as \( g^{-3} = \frac{1}{g^3} \) and \( 2g^{4} \) as it is, but it doesn’t simplify to \( 8g \). It would yield \( \frac{8g^{4}}{g^{3}} = 8g^{1} = 8g \), which means this is equivalent after simplification.
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4×2×g×g×g×gg×g×g
- This seems to be another representation but is not clear as written. Simplifying correctly, \( g^2 \) (not written as \( gg \)) added to \( g^3 \) does make sense, and it leads us to \( 8g \) equivalently.
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4×2×g×g×g×gg×g×g
- This is the same as the previous option and has a similar simplification.
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8g^7
- This does not match; \( g^7 \) is not equivalent to \( g^{1} \).
Thus, the equivalent expressions for \( (4g^{-3})(2g^4) = 8g \) are:
- \( 8g \)
- \( 4×g×g×g×2×g×g×g×g \)
- \( 4×2×g×g×g×gg×g×g \)
- and its variations that align with the above representations.
Final equivalent labels:
- \( 8g \)
- \( 4×g×g×g×2×g×g×g×g \) after simplification
- \( 4×2×g×g×g×gg×g×g \) after correct simplification
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