Practice Maths

L38 — Index Notation

Key Terms

index notation
A shorthand for repeated multiplication. E.g. 2 × 2 × 2 × 2 is written as 24. Also called exponential notation.
base
The number being multiplied repeatedly. In 24, the base is 2.
exponent (index)
The small raised number showing how many times the base is multiplied by itself. In 24, the exponent is 4. Read as “2 to the power of 4”.
power
The result of raising a base to an exponent. E.g. 24 = 16 is called “2 to the 4th power” or simply “the 4th power of 2”.

Base and Exponent Notation

  • The base is the number being multiplied.
  • The exponent tells us how many times the base is multiplied by itself.
  • We write: 24 and say “2 to the power of 4”.
  • 24 = 2 × 2 × 2 × 2 = 16

Special cases:

  • Any number to the power of 0 equals 1: n0 = 1 (e.g. 50 = 1)
  • Any number to the power of 1 equals itself: n1 = n (e.g. 71 = 7)
Hot Tip: 24 ≠ 2 × 4 = 8. This is a very common mistake! 24 means 2 multiplied by itself 4 times: 2 × 2 × 2 × 2 = 16.

Worked Examples

Example 1: Write 3 × 3 × 3 × 3 × 3 in index notation.

The base is 3, and it appears 5 times. Answer: 35

Example 2: Evaluate 53.

53 = 5 × 5 × 5 = 25 × 5 = 125

Example 3: Evaluate 104.

104 = 10 × 10 × 10 × 10 = 10 000

A Shorthand for Repeated Multiplication

Imagine you need to write “2 multiplied by itself 10 times.” In longhand: 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2. That’s tedious and error-prone. Index notation gives us a shortcut: 210. Much cleaner!

The base (2) is what’s being multiplied. The exponent (10) is how many times. Read it as “2 to the power of 10.” The result is 1024.

The Most Important Mistake to Avoid

24 does not equal 2 × 4 = 8. The exponent means “multiply the base by itself that many times” — not “multiply by the exponent.”

  • 24 = 2 × 2 × 2 × 2 = 16  —  NOT 8
  • 32 = 3 × 3 = 9  —  NOT 6
Remember: 24 ≠ 2 × 4. Always expand the base that many times: 24 = 2 × 2 × 2 × 2 = 16. If unsure, write out the expansion first, then multiply step by step.

Special Exponents: Zero and One

Two special rules are worth memorising:

  • Any number to the power of 1 equals itself: n1 = n.
  • Any number to the power of 0 equals 1: n0 = 1. This follows from the pattern of dividing by the base each time: 23 = 8, 22 = 4, 21 = 2, 20 = 1. Each step divides by 2.

Powers of 2 and Why Computers Use Them

Computers work in binary — they only know 0 and 1. All computer memory is measured in powers of 2:

  • 1 kilobyte = 210 = 1024 bytes
  • 1 megabyte = 220 ≈ 1 000 000 bytes
  • 1 gigabyte = 230 ≈ 1 000 000 000 bytes

The doubling pattern also appears in biology (bacteria doubling every hour), finance (compound interest), and many other real-world contexts.

Did You Know? The word “squared” comes from geometry. 32 = 9 because you can arrange 9 dots in a 3 × 3 square. Similarly, 33 = 27 is called “3 cubed” because 27 small cubes fit in a 3 × 3 × 3 big cube. The names connect maths to the physical world!

  1. Write in Index Notation

    Write each repeated multiplication in index notation.

    1. 4 × 4 × 4
    2. 7 × 7
    3. 2 × 2 × 2 × 2 × 2 × 2
    4. 10 × 10 × 10
    5. 5 × 5 × 5 × 5
    6. 6 × 6
    7. 3 × 3 × 3 × 3
    8. 9 × 9 × 9

  2. Evaluate Powers

    Calculate the value of each expression.

    1. 23
    2. 32
    3. 43
    4. 52
    5. 103
    6. 25
    7. 62
    8. 33

  3. Write as Repeated Multiplication

    Expand each index expression as a repeated multiplication (do not evaluate).

    1. 54
    2. 35
    3. 27
    4. 63
    5. 104
    6. 45
    7. 82
    8. 93

  4. Compare and Order Powers

    1. Which is larger: 23 or 32? Show working.
    2. Which is larger: 25 or 52? Show working.
    3. True or False: 24 = 42. Explain.
    4. True or False: 34 = 43. Explain.
    5. Arrange in order from smallest to largest: 24, 42, 33, 25.

  5. Patterns with Powers

    1. Complete the pattern: 21 = 2, 22 = 4, 23 = 8, 24 = __, 25 = __, 26 = __. What do you notice about consecutive terms?
    2. Without calculating, predict 210. Check by continuing the pattern.
    3. Complete the pattern for powers of 10: 101 = 10, 102 = 100, 103 = __, 104 = __, 105 = __.
    4. What is the connection between 10n and the number of zeros?

  6. Problem Solving with Powers

    1. A bacterium doubles every hour. Starting with 1 bacterium, write an expression for the number after 6 hours. Calculate the result.
    2. A computer has 210 kilobytes of storage. How many kilobytes is that?
    3. A square has a side length of 7 cm. Write the area as a power, then calculate it.
    4. A cube has a side length of 4 cm. Write the volume as a power, then calculate it.
    5. Which is greater: 28 or 82? By how much?

  7. Zero and One as Exponents

    Evaluate each expression. Use the rules n0 = 1 and n1 = n.

    1. 60
    2. 100
    3. 990
    4. 51
    5. 131
    6. 40 + 31
    7. 71 × 20
    8. 50 + 51 + 52

  8. Mixed Index Expressions

    Evaluate each expression. Follow the order of operations.

    1. 23 + 32
    2. 42 − 23
    3. 24 × 3
    4. 52 + 42
    5. 33 − 42
    6. 2 × 33
    7. (2 + 3)2
    8. 22 + 23 + 24

  9. Index Notation — Composite Expressions

    1. A square has side length 9 cm. Write the area as an index expression and evaluate it.
    2. A cube has side length 5 cm. Write the volume as an index expression and evaluate it.
    3. A school has 3 buildings, each with 3 floors, each with 3 classrooms, each with 3 rows of desks. Write the total number of rows as a power of 3, then calculate it.
    4. Write 1 000 000 as a power of 10. How many digits does the number have, and how does that relate to the exponent?
    5. A number written as 2n is equal to 128. What is the value of n? Show how you worked it out.

  10. Index Notation in Science and Technology

    1. A computer memory card can hold 28 files. How many files is that?
    2. In a phone tree, one person calls 3 others, each of whom calls 3 others, for a total of 4 rounds. How many people receive calls in the 4th round? Write your answer as a power.
    3. A lab experiment starts with 5 bacteria. They triple every hour. Write the number of bacteria after 4 hours as a power expression, then calculate it.
    4. The distance from Earth to the Sun is approximately 1.5 × 108 km. Write 108 as a regular number. Then write the full distance without scientific notation.
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