Index Notation

The little number counts the copies.

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 2³.

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. 2³ is read "2 to the power of 3", and a number written this way is called a power. The eight 2s above are 2⁸.

2·2·22³

Three copies of 2, multiplied together, are written 2³.

Squared and cubed

5² means 5 × 5 = 25. Lay 25 tiles in 5 rows of 5 and they make a square with side 5. That is why 5² is read "5 squared".

In the same way, 2³ = 2 × 2 × 2 = 8, and 8 small cubes stack into a cube 2 long, 2 wide and 2 high. That is why 2³ is read "2 cubed". Higher powers have no special name: 2⁴ is read "2 to the power of 4".

5 rows of 5 tiles make a square with side 5, so 5² = 25.

222

A cube with every edge 2 holds 2 × 2 × 2 = 8 small cubes, so 2³ = 8.

The index is not a multiplier

The index counts copies. It does not multiply the base: 2³ is 2 × 2 × 2 = 8, not 2 × 3 = 6. It does not add to the base either: 2³ is not 2 + 3 = 5.

The difference grows quickly. 3 × 4 is only 12, but 3⁴ = 3 × 3 × 3 × 3 = 81. Work a power out one multiplication at a time: 3 × 3 = 9, then 9 × 3 = 27, then 27 × 3 = 81.

2·2·22 × 2 × 2 = 8, not 6

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 2¹ = 2, 2² = 4, 2³ = 8, 2⁴ = 16, 2⁵ = 32: each is double the one before. So an amount that doubles again and again is written with a power of 2.

Worked example: A Colony of Bacteria That Doubles Every Hour

Question A 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. 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.

    550 h5 × 21101 h× 25 × 22202 h× 2after n hours: 5 × 2ncells
    550 h5 × 21101 h× 25 × 22202 h× 2after n hours: 5 × 2ncells
    Each hour multiplies the number of cells by one more 2: after n hours there are 5 × 2n cells.

  2. 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.

    550 h5 × 21101 h× 25 × 22202 h× 25 × 23403 h× 25 × 24804 h× 25 × 251605 h× 25 × 263206 h× 25 × 26= 5 × 64 = 320 cells
    550 h5 × 21101 h× 25 × 22202 h× 25 × 23403 h× 25 × 24804 h5 × 251605 h× 25 × 263206 h× 25 × 26= 5 × 64 = 320 cells
    (a) After 6 hours there are 5 × 26 = 5 × 64 = 320 cells.

  3. 3.For part (b) the number of cells must be more than 5000: 5 × 2n > 5000. Divide both sides by 5: 2n > 1000.

    550 h5 × 21101 h× 25 × 22202 h× 25 × 23403 h× 25 × 24804 h× 25 × 251605 h× 25 × 263206 h× 25 × 2n> 5000, so 2n> 1000
    550 h5 × 21101 h× 25 × 22202 h× 25 × 23403 h× 25 × 24804 h5 × 251605 h× 25 × 263206 h× 25 × 2n> 5000, so 2n> 1000
    More than 5000 cells means 5 × 2n > 5000, which is 2n > 1000.

  4. 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.

    550 h5 × 21101 h× 25 × 22202 h× 25 × 23403 h× 25 × 24804 h× 25 × 251605 h× 25 × 263206 h× 25 × 2n> 5000, so 2n> 100028= 25629= 512210= 1024512 < 1000 < 1024
    550 h5 × 21101 h× 25 × 22202 h× 25 × 23403 h× 25 × 24804 h5 × 251605 h× 25 × 263206 h× 25 × 2n> 5000, so 2n> 100028= 25629= 512210= 1024512 < 1000 < 1024
    29 = 512 is less than 1000, and 210 = 1024 is the first power of 2 that is more.

  5. 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.

    550 h5 × 21101 h× 25 × 22202 h× 25 × 23403 h× 25 × 24804 h× 25 × 251605 h× 25 × 263206 h× 25 × 2n> 5000, so 2n> 100028= 25629= 512210= 1024512 < 1000 < 1024after 10 h: 5 × 1024 = 5120 cells
    550 h5 × 21101 h× 25 × 22202 h× 25 × 23403 h× 25 × 24804 h5 × 251605 h× 25 × 263206 h× 25 × 2n> 5000, so 2n> 100028= 25629= 512210= 1024512 < 1000 < 1024after 10 h: 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

Common mistakes

  • 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.

More powers and roots problems, worked step by step →

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