EVALUATE NUMERICAL EXPRESSIONS INVOLVING WHOLE NUMBERS SHORTCUTS

Evaluate Numerical Expressions Involving Whole Numbers Shortcuts :

In this section, you will learn how to evaluate numerical expressions involving whole numbers.

Evaluating Expressions Involving Whole Number Using Algebraic Identities

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.

After having gone through the stuff given above, we hope that the students would have understood how to evaluate numerical expression involving whole numbers.

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