Demonstrate and verify the distributive property a(b+c) = ab + ac with numbers. Expand expressions, see step-by-step verification, area model visual, and comparison table for multiple examples.
The distributive property is one of the most fundamental rules in algebra: a × (b + c) = a × b + a × c. It allows you to "distribute" multiplication over addition (or subtraction). This property is the basis for expanding expressions, factoring, mental math shortcuts, and the FOIL method. For example, 7 × 13 can be computed as 7 × (10 + 3) = 70 + 21 = 91. This calculator lets you enter values for a, b, and c to verify the property numerically. It shows both sides of the equation, the step-by-step expansion, and an area model visualization where a(b + c) is shown as a rectangle of height a and width (b + c), split into two subrectangles of areas ab and ac. You can also toggle subtraction mode to verify a(b − c) = ab − ac, and explore multiple examples with presets. A comparison table computes both sides for a range of values so you can see the property holds universally.
The distributive property is so fundamental that students must internalize it before tackling polynomial algebra, factoring, and mental math shortcuts. This calculator verifies the identity numerically for any values you enter, shows the step-by-step expansion, draws an area model for geometric intuition, and demonstrates with a 10-row comparison table that the property holds universally — including with negatives, decimals, and three-term expressions.
a(b + c) = ab + ac a(b − c) = ab − ac Left side: a × (b + c) Right side: (a × b) + (a × c)
Result: 7(10 + 3) = 7×10 + 7×3 = 70 + 21 = 91
Both sides equal 91, verifying that 7(10 + 3) = 7×10 + 7×3. The area model shows a 7×13 rectangle split into 7×10 and 7×3.
The distributive property is the secret behind many mental math tricks. To compute 7 × 13 mentally, rewrite it as 7 × (10 + 3) = 70 + 21 = 91. To compute 8 × 99, write 8 × (100 − 1) = 800 − 8 = 792. These decompositions work because multiplication distributes over addition and subtraction. Practicing with this calculator builds the intuition to spot useful decompositions quickly.
The area model provides a visual proof: a rectangle with height a and width (b + c) can be split into two rectangles with areas ab and ac. The total area is a(b + c) = ab + ac. This geometric interpretation extends to algebra tiles for polynomial multiplication: (x + 2)(x + 3) is a rectangle of area x² + 5x + 6, split into four sub-rectangles (x², 2x, 3x, 6). The area model is the foundation of the FOIL method and polynomial long multiplication.
The distributive property generalizes to any number of terms: a(b + c + d) = ab + ac + ad. This is used in polynomial expansion, matrix-vector multiplication (distributing a scalar over a sum of vectors), and sigma notation in calculus. The extended distributive property is also the basis for expanding products of sums like (a + b)(c + d) = ac + ad + bc + bd, which itself follows from applying distribution twice.
It states that multiplication distributes over addition: a(b + c) = ab + ac. This means you can break a product into simpler parts.
Yes: a(b − c) = ab − ac. The distributive property applies to both addition and subtraction.
It is the foundation for expanding expressions, factoring polynomials, mental math, and the FOIL method for multiplying binomials. Use this as a practical reminder before finalizing the result.
A visual representation where a(b + c) is a rectangle of height a and width b + c, split into two rectangles with areas ab and ac. Keep this note short and outcome-focused for reuse.
Absolutely. For example, 3(5 + (−2)) = 3(3) = 9, and 3×5 + 3×(−2) = 15 − 6 = 9.
It extends to any number of terms: a(b + c + d) = ab + ac + ad. This is used in polynomial multiplication.