Simplify Expression: 9991 × 10009

The problem asks us to find the product of 9991 and 10009. We can simplify this calculation using an algebraic identity.

Using Algebraic Identity for Simplification

Notice that the numbers 9991 and 10009 are close to 10000. We can rewrite them as:

• \(9991 = 10000 - 9\)
• \(10009 = 10000 + 9\)

Now, the multiplication becomes:

\(9991 \times 10009 = (10000 - 9) \times (10000 + 9)\)

This is in the form of the difference of squares identity: \( (a - b)(a + b) = a^2 - b^2 \). Here, \(a = 10000\) and \(b = 9\).

Applying the Identity

Using the identity, we get:

\( (10000 - 9)(10000 + 9) = 10000^2 - 9^2 \)

Calculation Steps

Now, we calculate the squares:

• \( 10000^2 = (10^4)^2 = 10^8 = 100,000,000 \)
• \( 9^2 = 81 \)

Substitute these values back into the expression:

\( 10000^2 - 9^2 = 100,000,000 - 81 \)

Final Result

Performing the subtraction:

\( 100,000,000 - 81 = 99,999,919 \)

Therefore, the simplified value of the expression \(9991 \times 10009\) is 99,999,919.