Find the least common multiple.

Question

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

Answer 1

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.

1. Factorizing the constants:

   • $80=8\times 10=2^4\times 2\times 5=2^4\times 5$
   • $40=4\times 10=2^2\times 2\times 5=2^3\times 5$

2. Identifying the variable parts:

   • The variable part for the first expression is $d^2$.
   • The variable part for the second expression is $d^4$.

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 $2^4$ (from $80$).
     • For $5$, the maximum power is $5^1$ (from both).
   • Thus, the LCM of the constants is: $$LCM(80,40)=2^4\times 5^1=80$$

4. 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: $$LCM(d^2,d^4)=d^4$$

5. 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:

$$\boxed{{\displaystyle 80d^4}}$$