The expression (𝑎^6−𝑏^6)equals... [A] (𝑎^3−𝑏^3)^2 [B] (𝑎^2−𝑏^2)^3 [C] (𝑎−𝑏)^6 [D] none of these?

Question

math.  please show workings please

Answer 1

The only working necessary is that (X + Y)^n has n+1 terms in the expansion.  None of those will be "like terms" that can be combined unless Y is a constant multiple of X.

Since a^6 - b^6 has only two terms, but choices [A] through [C] all have 3 or more terms, then none of them can be equivalent.

Answer 2

A = (a³ - b³)²

A = (a³)² - 2a³b³ + (b³)²

A = a⁶ - 2a³b³ + b⁶

A ≠ a⁶ - b⁶

B = (a² - b²)³

B = (a² - b²)².(a² - b)

B = (a⁴ - 2a²b² + b⁴).(a² - b)

B = a⁶ - a⁴b - 2a⁴b² + 2a²b³ + a²b⁴ - b⁵

B ≠ a⁶ - b⁶

C = (a - b)⁶

C = [(a - b)²]³

C = [a² - 2ab + b²]³

C = [a² - 2ab + b²]².[a² - 2ab + b²]

C = [a⁴ - 2a³b + a²b² - 2a³b + 4a²b² - 2ab³ + a²b² - 2ab³ + b⁴].[a² - 2ab + b²]

C = [a⁴ + 6a²b² - 4a³b - 4ab³ + b⁴].[a² - 2ab + b²]

C = a⁶ - 2a⁵b + a⁴b² + 6a⁴b² - 12a³b³ + 6a²b⁴ - 4a⁵b + 8a⁴b² - 4a³b³ - 4a³b³ + 8a²b⁴ - 4ab⁵ + a²b⁴ - 2ab⁵ + b⁶

C = a⁶ - 6a⁵b + 15a⁴b² - 20a³b³ + 15a²b⁴ - 6ab⁵ + b⁶

C ≠ a⁶ - b⁶

…to you to conclude