Find the least common multiple.

80d2, 40d4
Write your answer as a constant times a product of single variables raised to exponents.

1 answer

To find the least common multiple (LCM) of the two expressions 80d2 and 40d4, we first break down each expression into its prime factors and variable components.

  1. Factorizing the constants:

    • 80=8×10=24×2×5=24×5
    • 40=4×10=22×2×5=23×5
  2. Identifying the variable parts:

    • The variable part for the first expression is d2.
    • The variable part for the second expression is d4.
  3. 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 24 (from 80).
      • For 5, the maximum power is 51 (from both).
    • Thus, the LCM of the constants is: LCM(80,40)=24×51=80
  4. Finding the LCM of the variable parts:

    • The LCM of the variable parts d2 and d4 is determined by taking the highest power:
      • The maximum power of d is d4.
    • Therefore, the LCM of the variable parts is: LCM(d2,d4)=d4
  5. Combining the results:

    • The overall LCM is the product of the LCM of the constants and the LCM of the variable parts: LCM(80d2,40d4)=80×d4

Thus, the least common multiple LCM(80d2,40d4) is:

80d4