EVALUATE NUMERICAL EXPRESSIONS INVOLVING WHOLE NUMBERS SHORTCUTS

Using the following algebraic identities, we may easily evaluate numerical expressions involving whole numbers.

(a + b)²  =  a² + 2ab + b²

(a - b)²  =  a² - 2ab + b²

a² - b²  =  (a + b) (a - b)

a³ + b³  =  (a + b) (a² - ab + b²)

a³ - b³  =  (a - b) (a² + ab + b²)

Example 1 :

Evaluate

796 ⋅ 796 - 204 ⋅ 204

Solution :

=  796 ⋅ 796 - 204 ⋅ 204

Instead of writing the same numerical values twice, we may write it once and take square for this.

=  796² - 204²

Now this exactly matches the algebraic identity 

a² - b²  =  (a + b) (a - b)

  =  (796 + 204) (796 - 204)

  =  1000 (592)

  =  592000

Example 2 :

Evaluate

387 ⋅ 387 + 113 ⋅ 113 + 2 ⋅ 387 ⋅ 113

Solution :

=  387 ⋅ 387 + 113 ⋅ 113 + 2 ⋅ 387 ⋅ 113

=  387² + 113² + 2 ⋅ 387 ⋅ 113

It is in the form a² + b² + 2ab, so we may write it as (a + b)²

  =  (387 + 113)²

  =  500²

  =  250000

Example 3 :

Evaluate 

87 ⋅  87 + 61 ⋅ 61 - 2 ⋅ 87 ⋅ 61

Solution :

 =  87 ⋅ 87 + 61 ⋅ 61 - 2 ⋅ 87 ⋅ 61

=  87² + 61² + 2 ⋅ 87 ⋅ 61

It is in the form a² + b² - 2ab, so we may write it as (a - b)²

  =  (87 - 61)²

  =  26²

  =  (25 + 1)²

=  25² + 1² + 2(25)(1)

=  625 + 1 + 50

=  676

Example 4 :

Evaluate

{(789 ⋅ 789 ⋅ 789 + 211 ⋅ 211 ⋅ 211)} / { (789 ⋅ 789 - 789 ⋅  211 + 211 ⋅ 211) }

Solution :

Let a  =  789 and b  =  211

Instead of whole numbers given in the question, let us use the variables a and b.

=  (a³ + b³) / (a² - ab + b²) 

=  (a + b)(a² - ab + b²) / (a² - ab + b²) 

=  (a + b)

by applying the values of a and b, we get

   =  789 + 211

  =  1000

Example 5 :

Evaluate

{(489 + 375)² - (489 - 375)²} / (489 ⋅ 375)

Solution :

Let a  =  489 and b  =  375

=  {(a + b)² - (a - b)²} / (a ⋅ b)

(a + b)²  =  (a² + 2ab + b²)   ---(1)

(a - b)²  =  (a² - 2ab + b²) ----(2)

(1) - (2)

By simplifying the numerator, we get

(a + b)²  - (a + b)²  =  (a² + 2ab + b²) -  (a² - 2ab + b²) 

=  4ab

By applying the value of (a + b)²  - (a + b)²

  =  4ab/ab

  =  4

Hence the answer is 4.

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