Evaluate 32^2 - 28^2.

Step-by-step solution

1. 1

   Spot the pattern before reaching for a calculator. The expression is literally one square minus another, which is the left side of the identity

   $a^{2}-b^{2}=(a-b)(a+b)$

   Here $a=32$ and $b=28$. Using the identity replaces two three-digit squarings ($1024$ and $784$) with one subtraction, one addition and one easy product.

2. 2

   Compute the two factors.

   $a-b=32-28=4,\qquad a+b=32+28=60$

   The difference is small because the two numbers are close together — which is exactly when this shortcut pays off most.

3. 3

   Multiply.

   $32^{2}-28^{2}=4\times60=240$

4. 4

   Verify the long way. $32^{2}=1024$ and $28^{2}=784$, so $1024-784=240$ ✓. Both routes were confirmed in exact integer arithmetic, so there is no rounding to worry about.

5. 5

   Note when the shortcut generalises. Any difference of two even squares $a^{2}-b^{2}$ with $a,b$ the same parity gives an even $a-b$ — here $4\times60$, a product that is easy to do mentally. The same trick turns $101^{2}-99^{2}$ into $2\times200=400$ instantly, whereas the direct route needs two four-digit squares.