Understand the distributive property and use it for faster multiplication, and discover algebraic identities for the square of a sum or difference.
The distributive property states a×(b+c) = a×b + a×c. This lets us break big multiplications into easier smaller ones, and is the basis for many mental math shortcuts.
Use the distributive property to find 15×104.
15×(100+4) = 15×100 + 15×4 = 1500+60 = 1560
Find 8×97 using distribution.
8×(100−3) = 800−24 = 776
(a+b)² = a²+2ab+b² and (a−b)² = a²−2ab+b². These identities let us square numbers close to round values quickly, without long multiplication.
Find 103² using the identity.
103² = (100+3)² = 100²+2×100×3+3² = 10,000+600+9 = 10,609
Find 98² using the identity.
98² = (100−2)² = 10,000−400+4 = 9,604
1. Use distribution to find 12×105.
2. Find 7×98 using distribution.
3. Find 102² using the identity for the square of a sum.
4. Find 97² using the identity for the square of a difference.
5. Expand (x+5)² using the identity.