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Simplify the following expression: 9991×10009
9999999
91999919
9999919
99999919
99999919
Express numbers close to the nearest power of 10. Here, 9991=10000−9 and 10009=10000+9. Apply the difference of squares formula: 100002−92.
Expression to simplify: 9991×10009
(a−b)(a+b)=a2−b2
Express numbers close to the nearest power of 10. Here, 9991=10000−9 and 10009=10000+9. Apply the difference of squares formula: 100002−92.
Multiplying directly or squaring incorrectly as 100002−9, forgetting to square the 9 to get 81.
Step 1: Express numbers in terms of a base
Notice that 9991 and 10009 are equidistant from 10000. We can write them as: 9991=10000−9 and 10009=10000+9.
(10000−9)×(10000+9)
Step 2: Apply the algebraic identity
Use the difference of squares identity (a−b)(a+b)=a2−b2 where a=10000 and b=9.
=100002−92
Step 3: Evaluate the squares and subtract
Calculate 100002=100,000,000 and 92=81. Subtract the two values.
=100000000−81=99999919
D is correct because 100002−92=100000000−81=99999919.
Recognizing numbers close to powers of 10 or 100 simplifies complex mental multiplications in quantitative aptitude exams.
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