S1 · Algebra · ALG.DISTRIBUTE

Apply distributive property

When distributing, every term inside the brackets is multiplied by the factor outside.

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Intuition

Understand first, compress later.

Imagine a rectangle of height a and width . Its area is , but also ab.

Switch representations

Multiplication as area14 · 12 = 168

10 · 104 · 1010 · 24 · 2

The rectangle is 14 wide and 12 tall, drawn to scale: its area is the product 14 · 12. Split the two sides into 10 + 4 and 10 + 2 and 4 sub-rectangles appear; their areas add back up to 168. That is exactly what the written method computes: every partial product is a piece of area.

Formal view

and . Signs belong to their terms.

Why does it work?

The distributive law links multiplication and addition. Both sides describe the same total of repeated groups.