Evaluate 501 × 501 Without Actual Multiplication — Using Algebraic Identity

501 × 501 = (501)²

Write 501 = 500 + 1

Using identity: (a + b)² = a² + 2ab + b² Here a = 500, b = 1

(500 + 1)² = 500² + 2 × 500 × 1 + 1² = 250000 + 1000 + 1 = 251001

Answer: 501 × 501 = 251001

Using (a + b)² = a² + 2ab + b²:

• 102 × 102: (100 + 2)² = 10000 + 400 + 4 = 10404

• 205 × 205: (200 + 5)² = 40000 + 2000 + 25 = 42025

Using (a − b)² = a² − 2ab + b²:

• 99 × 99: (100 − 1)² = 10000 − 200 + 1 = 9801

• 498 × 498: (500 − 2)² = 250000 − 2000 + 4 = 248004

Using (a + b)(a − b) = a² − b²:

• 501 × 499 = (500+1)(500−1) = 500² − 1 = 249999

• 103 × 97 = (100+3)(100−3) = 10000 − 9 = 9991

Three main identities for quick multiplication:

1. (a + b)² = a² + 2ab + b² Use when both numbers are just above a round number. Example: 501×501 = (500+1)²

2. (a − b)² = a² − 2ab + b² Use when both numbers are just below a round number. Example: 99×99 = (100−1)²

3. (a + b)(a − b) = a² − b² Use when one number is above and one below the same round number. Example: 501×499 = (500+1)(500−1) = 500²−1²