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Simplify the following expression: 9992×10008
91999964
99999936
9999964
9999936
99999936
Express the numbers close to a round number (10000): 9992=10000−8 and 10008=10000+8. Apply the algebraic identity (a−b)(a+b)=a2−b2 to compute the product in seconds.
Expression to simplify: 9992×10008
(a−b)(a+b)=a2−b2
Express the numbers close to a round number (10000): 9992=10000−8 and 10008=10000+8. Apply the algebraic identity (a−b)(a+b)=a2−b2 to compute the product in seconds.
Multiplying the two large numbers directly, which takes a lot of time and increases the risk of calculation errors in exam conditions.
Step 1: Express numbers in terms of a base
Rewrite 9992 as (10000−8) and 10008 as (10000+8).
9992×10008=(10000−8)(10000+8)
Step 2: Apply Algebraic Identity
Use the difference of squares formula (a−b)(a+b)=a2−b2, where a=10000 and b=8.
(10000−8)(10000+8)=100002−82
Step 3: Calculate Squares
Compute 100002=100000000 and 82=64.
100000000−64
Step 4: Final Subtraction
Subtract 64 from 100000000 to get the final simplified result.
99999936
B is correct because applying the algebraic identity (10000−8)(10000+8)=100002−82=100000000−64=99999936.
This identity (a−b)(a+b)=a2−b2 is extremely useful in quantitative aptitude for mental math, simplification, and mensuration problems.
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