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Simplify the following expression: 9993×10007
91999949
9999951
9999951
9999949
9999951
Use the identity (a−b)(a+b)=a2−b2 where a=10000 and b=7. Simply calculate 100002−72=100,000,000−49=99,999,951.
We are given the expression 9993 multiplied by 10007.
(a−b)(a+b)=a2−b2
Use the identity (a−b)(a+b)=a2−b2 where a=10000 and b=7. Simply calculate 100002−72=100,000,000−49=99,999,951.
Students often try to perform long multiplication, which is time-consuming and prone to manual calculation errors, rather than using the identity.
Identify the values
Rewrite the expression 9993×10007 by expressing both numbers relative to a common base of 10000.
9993=(10000−7),10007=(10000+7)
Apply algebraic identity
Substitute these into the difference of squares identity (a−b)(a+b)=a2−b2.
(10000−7)(10000+7)=100002−72
Calculate the squares
Calculate 100002=100,000,000 and 72=49.
100,000,000−49=99,999,951
C is correct because the product of 9993 and 10007 calculated via the difference of squares identity is 99,999,951.
This identity is a fundamental shortcut in speed math; applying it to numbers close to powers of 10 (like 99 or 101) saves significant time in exams.
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