Multiplying fractions and powers of a fraction — Mathematics, 11–13
A fraction of a fraction is a product, and repeating the same fraction as a factor gives a power such as (2/3)³.
A fraction of a fraction
Multiplying fractions means taking a fraction of a fraction. 1/2 × 1/5 is half of a fifth. Draw a square of side 10 with 100 cells: 1/5 is 20 cells, half of that is 10 cells, that is 10/100 = 1/10. So you multiply the numerators (1 × 1) and the denominators (2 × 5): 1/10. In general, multiply top by top and bottom by bottom, and shorten at the end: 3/4 × 2/9 = 6/36 = 1/6.
Why the denominators multiply
Why is the product of two fractions below 1 smaller than each of them? Because a fraction of a fraction is only a piece of a piece: half of 1/5 is less than 1/5, and 1/10 is smaller than both factors. Another way to see it, with the rectangle model: cut a unit square into 3 equal rows and 5 equal columns. Every cell is 1/3 × 1/5 = 1/15 of the square, and 15 cells fill it. That is why the denominators multiply.
2/3 × 4/5 and (2/3)³
2/3 × 4/5. Split a unit square into 3 rows and 5 columns: 15 cells. Shade 2 rows (2/3) and 4 columns (4/5). The cells shaded both ways form a 2 by 4 rectangle: 8 cells out of 15, so 2/3 × 4/5 = 8/15. A power repeats the same factor: (2/3)³ = 2/3 × 2/3 × 2/3. The numerators give 2 × 2 × 2 = 8 and the denominators 3 × 3 × 3 = 27, so (2/3)³ = 8/27.
The exponent belongs to both numbers
In (2/5)², many people square only one of the numbers, or treat the exponent as a multiplier: 2² = 4 over 5 gives 4/5, and so does 2/5 × 2 = 4/5. Both are wrong. The power (2/5)² means 2/5 × 2/5, and the exponent says how many times the whole fraction is a factor, so it acts on the numerator and on the denominator: 2²/5² = 4/25. A check: 4/5 is greater than 2/5, but a product of two fractions below 1 must be smaller than each factor.
Recipes and folded paper
A recipe asks for 3/4 of a cup of flour and you make only 2/3 of the recipe: 2/3 × 3/4 = 6/12 = 1/2 cup. When you fold a sheet in half again and again, each fold halves the area: after 3 folds each part is (1/2)³ = 1/8 of the sheet, that is 0.125, and after 5 folds it is (1/2)⁵ = 1/32. In your head you can go on: after 10 halvings the part is (1/2)¹⁰ = 1/1024, exactly one cell of a square grid of side 32.
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