Question

Factorize (256⁴-1).
Use appropriate identity.

Answer 1

(256⁴-1)

= (256-1)⁴

Using identity (a-b)⁴ = a⁴−4a³b+6a²b²−4ab³+b⁴

Let a be 256 and b be 1

Then

256⁴−4(256)³(1)+6(256)²(1)²−4(256)(1)³+(1)⁴

After solving

(256²-1)²

(a-b)² = a²-2ab+b²

256²-2×256×1+1²

= (256²-1)(256²+1)

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Answer 2

Answer:

Use identity:

• a² - b² = (a + b)(a - b)

Consider that:

• 256 = 2⁸

Now factorize:

• 256⁴ - 1 =
• (2⁸)⁴ - 1 =
• 2³² - 1 =
• (2¹⁶ - 1)(2¹⁶ + 1) =
• (2⁸ - 1)(2⁸ + 1)(2¹⁶ + 1) =
• (2⁴ - 1)(2⁴ + 1)(2⁸ + 1)(2¹⁶ + 1) =
• (2² - 1)(2² + 1)(2⁴ + 1)(2⁸ + 1)(2¹⁶ + 1) =
• (2 - 1)(2 + 1)(2² + 1)(2⁴ + 1)(2⁸ + 1)(2¹⁶ + 1)