To find the least common multiple (LCM) of the two expressions \(80d^2\) and \(40d^4\), we first break down each expression into its prime factors and variable components.
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Factorizing the constants:
- \(80 = 8 \times 10 = 2^4 \times 2 \times 5 = 2^4 \times 5\)
- \(40 = 4 \times 10 = 2^2 \times 2 \times 5 = 2^3 \times 5\)
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Identifying the variable parts:
- The variable part for the first expression is \(d^2\).
- The variable part for the second expression is \(d^4\).
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Finding the LCM of the constants:
- The LCM of the constants \(80\) and \(40\) can be found by taking the highest power of each prime factor:
- For \(2\), the maximum power is \(2^4\) (from \(80\)).
- For \(5\), the maximum power is \(5^1\) (from both).
- Thus, the LCM of the constants is: \[ LCM(80, 40) = 2^4 \times 5^1 = 80 \]
- The LCM of the constants \(80\) and \(40\) can be found by taking the highest power of each prime factor:
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Finding the LCM of the variable parts:
- The LCM of the variable parts \(d^2\) and \(d^4\) is determined by taking the highest power:
- The maximum power of \(d\) is \(d^4\).
- Therefore, the LCM of the variable parts is: \[ LCM(d^2, d^4) = d^4 \]
- The LCM of the variable parts \(d^2\) and \(d^4\) is determined by taking the highest power:
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Combining the results:
- The overall LCM is the product of the LCM of the constants and the LCM of the variable parts: \[ LCM(80d^2, 40d^4) = 80 \times d^4 \]
Thus, the least common multiple \(LCM(80d^2, 40d^4)\) is:
\[ \boxed{80d^4} \]