Question

Find the value of the expression: \[\dfrac{{{{\left( {398 + 257} \right)}^2} - {{\left( {398 - 257} \right)}^2}}}{{398 \times 257}}\].

Answer 1

Hint: Here we will use basic formula of algebra and basic algebraic identity\[{\left( {\text{a}} \right)^2} - {\left( {\text{b}} \right)^2} = \left( {{\text{a}} + {\text{b}}} \right)\left( {{\text{a}} - b} \right)\].
Algebraic identities help us in converting complex problems into more simplified problems. Some of them are\[{\left( {{\text{a}} + {\text{b}}} \right)^2} - {\left( {{\text{a}} - {\text{b}}} \right)^2}4{\text{ab}}\], \[{{\text{a}}^3} + {{\text{b}}^3} = \left( {{\text{a}} + {\text{b}}} \right)\left( {{{\text{a}}^2} + {{\text{b}}^2} - {\text{ab}}} \right)\]etc.

Complete step-by-step answer:
We observe that given expression can be converted to a very famous identity by simple substitution of the numbers \[398 + 257\] and \[398 - 257\]. Let the number \[398 + 257\] is represented by variable “a” and number \[398 - 257\] is represented by variable “b”.
Substituting the variables in the given expression we get, \[\dfrac{{{{\left( {\text{a}} \right)}^2} - {{\left( {\text{b}} \right)}^2}}}{{398 \times 257}}\]. Now, expanding the numerator using the following identity\[{\left( {\text{a}} \right)^2} - {\left( {\text{b}} \right)^2} = \left( {{\text{a}} + {\text{b}}} \right)\left( {{\text{a}} - {\text{b}}} \right)\], expansion is given as \[\dfrac{{\left( {{\text{a}} + {\text{b}}} \right)\left( {{\text{a}} - {\text{b}}} \right)}}{{398 \times 257}}\].
To solve it further we will back substitute the respective values of variables a and b, we obtain,
$\Rightarrow$\[\dfrac{{\left( {398 + 257 + 398 - 257} \right)\left( {398 + 257 - 398 + 257} \right)}}{{398 \times 257}}\].
As in the above expression we observe that numbers are of opposite signs therefore, simplifying this expression by cancelling such numbers, we get
$\Rightarrow$\[\dfrac{{\left( {2 \times 398} \right)\left( {2 \times 257} \right)}}{{398 \times 257}}\] .
Simplifying the above expression further we get \[\dfrac{{\left( {2 \times 398} \right)\left( {2 \times 257} \right)}}{{398 \times 257}} = 4\].
Therefore the value of the expression \[\dfrac{{{{\left( {398 + 257} \right)}^2} - {{\left( {398 - 257} \right)}^2}}}{{398 \times 257}}\] is equal to \[4\].

4 is the value of the given algebraic expression.

Note: In such types of problems where, we must try to find the best suitable identity. For example, in the given problem it was easy to identify that identity \[{\left( {\text{a}} \right)^2} - {\left( {\text{b}} \right)^2} = \left( {{\text{a}} + {\text{b}}} \right)\left( {{\text{a}} - b} \right)\] can be used by simple substitution. If we observe that a number or numbers are repeated two or more times in an expression then we must start substituting those variables. After substituting we should try to observe whether the substituted expression matches with some known identity or not, if it doesn’t match then we must try to alter the substitution.