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Evaluate the following using identities:
(a) $1007 \times 1004$
(b) $121^{2}-31^{2}$
Given:
The given expressions are (a) $1007 \times 1004$ and (b) $121^{2}-31^{2}$.
To do :
We have to evaluate the given expressions using identities.
Solution :
(a) $1007 \times 1004$
We know that,
$(a+b) \times (c+d) = a \times (c+d) + b \times (c+d)$
Therefore,
$1007 \times 1004 = (1000+7) \times (1000+4)$
$ = 1000 \times 1000 + 1000 \times 4 + 7 \times 1000 + 7 \times 4$
$= 1000000 + 4000 + 7000 + 28$
$ = 1011028$.
The value of $1007 \times 1004$ is $1011028$.
(b) $121^{2}-31^{2}$
We know that,
$a^2 - b^2 = (a+b) (a-b)$
Therefore,
$121^{2}-31^{2} = (121+31)(121-31)$
$= (152)(90)$
$= 13680$
The value of $121^{2}-31^{2}$ is $13680$.