Let’s find the product by formulae;

i.

ii.

iii.

iv.

v.

_{Formula used:}

_{(a + b)(a – b) = a}² – b²

_{i. (xy + pq)(xy – pq)}

Taking a = xy and b = pq, then from above identity,

= (xy)² – (pq)²

= x²y² – p²q²

ii. 49 × 51

= (50 – 1)(50 + 1)

Taking a = 50 and b = 1, then from above identity,

= 50² - 1²

= 2500 – 1

= 2499

iii. (2x – y + 3z)(2x + y + 3z)

Taking a = 2x and b = y + 3z, the from the above identity we have

= (2x)² – (y + 3z)²

= 4x² – (y² + 2(y)(3z) + (3z)²) [∵ (a + b)² = a² + 2ab + b²]

= 4x² – (y² + 6yz + 9z²)

= 4x² – y² – 6yz – 9z²

iv. 1511 × 1489

= (1500 + 11)(1500 – 11)

Taking a = 1500 and b = 11, then from above identity,

= (1500)² - 11²

= 2250000 – 121

= 2249879

v. (a – 2)(a + 2)(a² + 4)

= (a² – 2²)(a² + 4) [Taking a = a, and b = 2 in above identity]

= (a² – 4)(a² + 4)

= (a²)² - 4² [Taking a = a² and b = 4 in above identity]

= a⁴ – 16