By solving problems of maths, you will be able to solve any problems in a matter of time.

LHS

(a+b)² 

= (a+b) (a+b)

= a² +ab +ba + b²

= a² + ab + ab + b²

= a² + 2ab + b²

By applying (a+b)² = a² + 2ab + b²

We can write 

1005²

= (1000 + 5)²

= 1000² + 2×1000×5 +5²

= 1000000 + 10000 + 25

= 1010025

Lhs

(a-b)² 

= (a-b) (a-b)

= a² - ab - ba + b²

= a² - ab - ab + b²

= a² - 2ab + b²

       Or

  a² + b² - 2ab

By applying (a-b)² = a² - 2ab + b²

We can write 

999²

= (1000 - 1)²

= 1000² - 2×1000×1 + 1²

= 1000000 - 2000 + 1

= 9 98 001 Ans.

LHS 

(a+b)³

It can be written as 

(a+b) (a+b)²

= (a+b) ( a² + b² + 2ab )

= a x a² + a x b² + a x 2ab + b x a² + b x b² + b x 2ab

= a³ + ab² + 2a²b  + ba² + b³ + 2ab²

= a³ + b³ + 3ab² + 3a²b

= a³ + b³ + 3ab (a+b) 

Hence proved 

By applying (a+b)³ = a³ + b³ + 3ab (a+b)

We can write 

(💯 +2)³ = 100³ + 2³ + 3 x 100 x 2 ( 100 + 2 )

= 100 x 100 x 100 + 2 x 2 x 2 + 600 ( 100 + 2 )

= 1000000 + 8 + 60000 + 1200

= 10 61 208 Ans.

Proof 🧾, LHS : (a-b)³

We can write ✍️ it as (a-b) (a-b)²

Now by applying (a-b)² = a² -2ab +b²

= (a-b) ( a² - 2ab + b² )

= a x a² - a x 2ab + a x b² - b x a² + b x 2ab - b x b²

= a³ - 2a²b + ab² - ba² + 2ab² - b³

= a³ - b³ - 3a²b + 3ab²

= a³ - b³ -3ab ( a - b )

= RHS 

Hence proved.

999³

We can write it as (1000-1)³

 Now, by applying

(a-b)³ = a³ - b³ - 3ab (a - b)

We can write ✍️

(1000 - 1)³ = 1000³ - 1³ - 3 x 1000 x 1 ( 1000 - 1 )

= 1000 x 1000 x 1000 - 1 - 3000 ( 1000 - 1 )

= 1000000000 - 1 - 3000000 + 3000

= 1000003000 - 3000001

= 997002999 Ans.

Proof

LHS : ( a + b + c )²

We can write ✍️ it as 

( a + b + c ) ( a + b + c )

= a x a + a x b + a x c + b x a + b x b + b x c + c x a + c x b + c x c 

= a² + ab + ac + ba + b² + bc + ca + cb + c²

Now, we know that

ab = ba, bc = cb and ac = ca 

Multiplication is commutative 

Which will be equal to

a² + b² + c² + 2ab + 2bc + 2ca = RHS

Hence proved.