An index counts how many copies of the base are multiplied together
Counting copies
Writing 2 × 2 × 2 is slow, and 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 is slower. Index notation writes a repeated multiplication in a short way: 2 × 2 × 2 is written .
The large number, 2, is the base: it is the number being multiplied. The small raised number, 3, is the index. It counts how many copies of the base are multiplied together. is read "2 to the power of 3", and a number written this way is called a power. The eight 2s above are .
Three copies of 2, multiplied together, are written .
Squared and cubed
means 5 × 5 = 25. Lay 25 tiles in 5 rows of 5 and they make a square with side 5. That is why is read "5 squared".
In the same way, , and 8 small cubes stack into a cube 2 long, 2 wide and 2 high. That is why is read "2 cubed". Higher powers have no special name: is read "2 to the power of 4".
5 rows of 5 tiles make a square with side 5, so .
A cube with every edge 2 holds 2 × 2 × 2 = 8 small cubes, so .
The index is not a multiplier
The index counts copies. It does not multiply the base: is 2 × 2 × 2 = 8, not 2 × 3 = 6. It does not add to the base either: is not 2 + 3 = 5.
The difference grows quickly. 3 × 4 is only 12, but . Work a power out one multiplication at a time: 3 × 3 = 9, then 9 × 3 = 27, then 27 × 3 = 81.
The index 3 asks for three copies of 2, multiplied, which make 8.
One more copy each time
Raising the index by 1 multiplies by the base once more. The powers of 2 run , , , , : each is double the one before. So an amount that doubles again and again is written with a power of 2.
Powers and roots · A Colony of Bacteria That Doubles Every Hour
QuestionA dish holds 5 bacteria cells at the start of an experiment. The number of cells doubles every hour. (a) Write the number of cells after 6 hours in index notation and find its value. (b) After how many whole hours does the number of cells first pass 5000?
1.Let n be the number of hours that have passed. After 1 hour there are 5 × 2 cells and after 2 hours there are 5 × 2 × 2 = 5 × 22 cells. Each hour multiplies by one more 2, so after n hours there are 5 × 2n cells.
Each hour multiplies the number of cells by one more 2: after n hours there are 5 × 2n cells. 2.(a) After 6 hours there are 5 × 26 cells. Since 26 = 2 × 2 × 2 × 2 × 2 × 2 = 64, this is 5 × 64 = 320 cells.
(a) After 6 hours there are 5 × 26 = 5 × 64 = 320 cells. 3.For part (b) the number of cells must be more than 5000: 5 × 2n > 5000. Divide both sides by 5: 2n > 1000.
More than 5000 cells means 5 × 2n > 5000, which is 2n > 1000. 4.List the powers of 2 near 1000: 28 = 256, 29 = 512 and 210 = 1024. The power 29 is less than 1000, and 210 is the first power of 2 that is more than 1000, so n = 10.
29 = 512 is less than 1000, and 210 = 1024 is the first power of 2 that is more. 5.(b) The number of cells first passes 5000 after 10 hours. Check: after 9 hours there are 5 × 512 = 2560 cells, which is less than 5000, and after 10 hours there are 5 × 1024 = 5120 cells.
(b) The number of cells first passes 5000 after 10 hours, when there are 5120 cells.
Answer: (a) 5 × 26 = 320 cells; (b) 10 hours
Watch out for
- Working out 5 × 2 × 6 = 60 for part (a). The index 6 does not multiply the 2. It counts how many times 2 is a factor, so 26 = 64 and not 12.
- Working out (5 × 2)6 = 106. Only the 2 is repeated each hour. The 5 cells at the start are counted once, so the number is 5 × 26.
Your turn
1.What is ?