Question

Find the LCM of ( a² - 4 ), ( a³ - 8 ), and ( (a+2)² ).

a. ( (a+2)³ )
b. ( a³ - 8 )
c. ( a³ + 2a² - 4a - 8 )
d. ( (a+2)²(a-2) )

Answer 1

The LCM of the expressions (a² - 4), (a³ - 8), and ((a+2)²) is ((a+2)²)(a - 2), which is option d.

Step-by-step explanation:

To find the LCM (Least Common Multiple) of the given expressions (a² - 4), (a³ - 8), and ((a+2)²), we have to factor each expression first:

• (a² - 4) can be factored as (a + 2)(a - 2) since it's a difference of squares.
• (a³ - 8) can be factored as (a - 2)(a² + 2a + 4), which is the difference of cubes.
• ((a+2)²) is already a perfect square.

Now, to find the LCM, we take the highest power of each factor that appears in any of the expressions:

• The highest power of (a - 2) is from (a³ - 8), which is (a - 2).
• The largest power of (a + 2) is from ((a+2)²), which is ((a+2)²).
• The factor (a² + 2a + 4) only appears in (a³ - 8), so it's included once.

Thus, the LCM is ((a+2)²)(a - 2). The correct answer is option d.