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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