Practice Maths

Establishing Index Notation

Key Terms

Index notation
A compact way to write repeated multiplication; e.g. 34 means 3 × 3 × 3 × 3.
Base
The number being multiplied repeatedly; in 34, the base is 3.
Index (exponent)
How many times the base is multiplied by itself; in 34, the index is 4.
Power
An expression written in index notation; e.g. 34 is read “three to the power of four.”

Reading Index Notation

We read 25 as “two to the power of five” or “two to the fifth.”
Special cases: 32 = “three squared” and 43 = “four cubed.”

Hot Tip The index tells you how many times the base is multiplied by itself — not how many times it is multiplied by something else. So 23 = 2 × 2 × 2 = 8, not 2 × 3 = 6.

Worked Example

Question: Write 4 × 4 × 4 × 4 × 4 in index notation, then evaluate.

Step 1 — Count how many times 4 is multiplied.
4 appears 5 times, so the index is 5.

Step 2 — Write in index notation.
4 × 4 × 4 × 4 × 4 = 45

Step 3 — Evaluate.
45 = 4 × 4 × 4 × 4 × 4 = 16 × 16 × 4 = 1024

What Is Index Notation and Why Do We Use It?

Imagine writing out 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 every time you wanted to describe a large repeated multiplication. Index notation gives us a shortcut. Instead, we write 210 — the base (2) tells us what is being multiplied, and the index (10) tells us how many times. This compact form is used everywhere from computer memory sizes to scientific measurements of atoms and galaxies.

In the expression 54, the base is 5 and the index (also called the exponent or power) is 4. This means: 5 × 5 × 5 × 5 = 625. We say it aloud as "five to the power of four" or "five to the fourth." The special cases are: squares (power of 2) and cubes (power of 3), which come from geometry — a square with side 5 has area 52, and a cube with side 5 has volume 53.

Evaluating Powers Step by Step

To evaluate 35, don't try to do it all at once. Build it up: 3 × 3 = 9, then 9 × 3 = 27, then 27 × 3 = 81, then 81 × 3 = 243. So 35 = 243. A common mistake is to multiply base × index (e.g. saying 35 = 15). This confuses repeated multiplication with ordinary multiplication — the index is a count, not a factor.

Powers of 10 are especially useful: 101 = 10, 102 = 100, 103 = 1000, 104 = 10 000. Notice the pattern: the index equals the number of zeros. This is exactly why our place value system works so well.

Key Tip: The index is a counter, not a multiplier. 23 means 2 × 2 × 2 = 8, not 2 × 3 = 6. Always expand it out if you're unsure — count how many 2s you're multiplying together.

The Zero Index and the Power of One

Two special rules often confuse students. First, any non-zero number to the power of zero equals 1. Here's why: consider 34 ÷ 34. This equals 1 (anything divided by itself). But using the division index law (which we'll cover soon), it also equals 34−4 = 30. So 30 = 1. Second, any number to the power of 1 equals itself: 71 = 7. This is because there's just one copy of the base.

A common misconception: students think 50 = 0. Think about it this way — 52 = 25, 51 = 5, 50 = 1. Each time we decrease the index by 1, we divide by 5. So 5 ÷ 5 = 1, confirming 50 = 1, not 0.

Key Tip: a0 = 1 for any non-zero value of a. It is never 0. Think of the decreasing pattern: as the index counts down toward zero, the value divides by the base, eventually reaching 1.

Writing Numbers Using Index Notation

To write 7 × 7 × 7 × 7 in index notation: count how many 7s appear (four of them), so the answer is 74. To reverse this: write 43 in expanded form as 4 × 4 × 4. Sometimes the base is not a whole number — you can have (0.5)4 or (1/2)4, which means 0.5 × 0.5 × 0.5 × 0.5 = 0.0625.

Real-World Connections

Index notation appears constantly in real life. Computer storage uses powers of 2: 1 kilobyte = 210 bytes = 1024 bytes. A megabyte = 220 bytes. Bacteria doubling every hour: after 8 hours you have 28 = 256 bacteria. In finance, compound interest uses powers to calculate growth over many years. Wherever things grow by repeated multiplication, index notation describes it efficiently.

Mastery Practice

  1. Write each repeated multiplication in index notation. Fluency

    1. 5 × 5 × 5 × 5
    2. 2 × 2 × 2 × 2 × 2 × 2
    3. 7 × 7 × 7
    4. 10 × 10 × 10 × 10 × 10
    5. 3 × 3
    6. 6 × 6 × 6 × 6
    7. 9 × 9 × 9 × 9 × 9
    8. 12 × 12

  2. Evaluate each expression (find the value). Fluency

    1. 23
    2. 34
    3. 102
    4. 53
    5. 42
    6. 26
    7. 62
    8. 103

  3. Write each in expanded (repeated multiplication) form. Fluency

    1. 54
    2. 35
    3. 73
    4. 27
    5. 104
    6. 82
    7. 43
    8. 64

  4. Evaluate each expression involving zero and one as the index. Understanding

    1. 70
    2. 151
    3. 1000
    4. (−3)0
    5. 50 + 21
    6. 41 × 30
    7. 100 + 101 + 102
    8. 60 × 61 × 62

  5. Arrange each set of numbers in ascending order (smallest to largest). Evaluate each power first. Understanding

    1. 25,   33,   52,   43
    2. 102,   73,   210,   54
    3. 63,   92,   44,   28
    4. 1100,   21,   30,   05

  6. Real-world index notation. Problem Solving

    1. A bacterium doubles every hour. Starting from 1 bacterium, write the number of bacteria after 6 hours in both index notation and as a standard number.
    2. A computer has storage measured in bytes. 1 kilobyte = 103 bytes, 1 megabyte = 106 bytes, 1 gigabyte = 109 bytes.
      1. How many bytes are in 1 megabyte? Write your answer as both a power of 10 and a standard number.
      2. How many kilobytes are in 1 gigabyte?
    3. A square tile has a side length of 4 cm. A larger square is made by arranging these tiles in a 4 × 4 grid.
      1. Write the area of one tile in index notation.
      2. Write the total area of the large square in index notation, then evaluate.
    4. Mia says “34 = 12 because 3 × 4 = 12.” Explain her mistake and give the correct answer.

  7. For each expression, state the base and the index. Fluency

    1. 75
    2. 29
    3. 106
    4. 43
    5. 81
    6. 152
    7. 67
    8. 1000

  8. For each, write the expression in index notation AND evaluate it. Fluency

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

  9. State whether each statement is True or False. Justify your answer with a calculation. Understanding

    1. 24 = 42
    2. 32 > 23
    3. 150 = 50
    4. 50 = 0
    5. 05 = 0
    6. 25 < 52
    7. 104 = 4 × 10

  10. Patterns and powers. Problem Solving

    1. Complete this table and describe the pattern you notice:
      1. 21 = ?
      2. 22 = ?
      3. 23 = ?
      4. 24 = ?
      5. 25 = ?
    2. A piece of paper is folded in half repeatedly. Each fold doubles the number of layers. Write the number of layers after 8 folds in index notation, then evaluate.
    3. Which is greater: 210 or 102? Show your working.
    4. Write three different expressions in index notation that all equal 64.
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