Practice Maths

Multiplication and Division Index Laws

Key Terms

Multiplication law
am × an = am+n — when multiplying powers with the same base, add the indices.
Division law
am ÷ an = am−n — when dividing powers with the same base, subtract the indices.
Same base
Both terms must share the same base for an index law to apply; the laws do not work when the bases differ.
Hot Tip Always check the bases match before applying an index law. If the bases are different, the law does not apply.

Worked Example

Multiplication: Simplify 34 × 35

Same base (3), so add the indices: 34+5 = 39

Division: Simplify 78 ÷ 73

Same base (7), so subtract the indices: 78−3 = 75

Why the Multiplication Law Works

Let's think carefully about why am × an = am+n. Consider 23 × 24. Written out fully: (2 × 2 × 2) × (2 × 2 × 2 × 2) = 2 × 2 × 2 × 2 × 2 × 2 × 2 = 27. We're just combining two groups of repeated multiplication into one, so the total count is m + n. This is NOT a coincidence — it follows directly from the definition of index notation.

The critical condition: the bases must be identical. You cannot combine 23 × 54 using this law because you have twos and fives — different things. It would be like counting apples and oranges as one group.

Why the Division Law Works

Similarly, am ÷ an = am−n because division cancels factors. Consider 37 ÷ 33 = (3 × 3 × 3 × 3 × 3 × 3 × 3) ÷ (3 × 3 × 3). Three of the 3s in the numerator cancel with the three in the denominator, leaving 34. So we subtracted the indices: 7 − 3 = 4. Again, bases must be the same.

An important special case: when m = n, we get am ÷ am = a0 = 1. This confirms the zero-index rule from the previous lesson! Any number divided by itself is 1, and that corresponds to a0 = 1.

Key Tip: Always check the bases match before applying an index law. 54 × 53 = 57 (same base, add). But 54 × 33 cannot be simplified — different bases mean the law does not apply.

Working with Combined Operations

When an expression involves both multiplication and division, work left to right, applying the laws step by step. For example: 25 × 23 ÷ 24. Step 1: 25 × 23 = 28. Step 2: 28 ÷ 24 = 24. Alternatively, combine all at once: add the indices for multiplication and subtract for division: 5 + 3 − 4 = 4, giving 24.

Multiplying by a0 doesn't change the result (since a0 = 1). So 75 × 70 = 75+0 = 75. This makes sense — multiplying by 1 changes nothing.

Key Tip: When simplifying, handle all multiplications and divisions together: add indices for ×, subtract indices for ÷. Keep track with a running total: 5 + 3 − 4 = 4, so the answer is the base to the power 4.

Connecting to Real-World Size

These laws are essential in science and computing. If a virus multiplies so that after 6 hours there are 106 copies, and in the next 4 hours they multiply again reaching 106 × 104, the total is 1010 — 10 billion viruses. The index laws let us do these calculations without writing out enormous numbers. In engineering, quantities involving areas (m2) and volumes (m3) use these laws constantly when converting between units.

Mastery Practice

  1. Apply the multiplication index law to simplify. Leave answers in index notation. Fluency

    1. 23 × 24
    2. 52 × 56
    3. 35 × 33
    4. 74 × 72
    5. 103 × 105
    6. 41 × 47
    7. 64 × 64
    8. 92 × 93

  2. Apply the division index law to simplify. Leave answers in index notation. Fluency

    1. 57 ÷ 53
    2. 29 ÷ 24
    3. 68 ÷ 62
    4. 106 ÷ 101
    5. 37 ÷ 33
    6. 75 ÷ 75
    7. 49 ÷ 44
    8. 86 ÷ 83

  3. Mixed multiplication and division. Simplify each expression. Fluency

    1. 23 × 25 ÷ 24
    2. 36 ÷ 32 × 31
    3. 54 × 52 ÷ 53
    4. 107 ÷ 103 ÷ 102
    5. 45 × 43 ÷ 46
    6. 72 × 74 × 71
    7. 68 ÷ 65 × 60
    8. 210 ÷ 26 × 22

  4. Find the missing index in each equation. Understanding

    1. 2□ × 23 = 27
    2. 56 ÷ 5□ = 52
    3. 32 × 3□ = 39
    4. 7□ ÷ 74 = 73
    5. 105 × 10□ = 105
    6. 4□ × 4□ = 48 (both indices are equal)
    7. 610 ÷ 6□ = 66
    8. 9□ ÷ 93 = 90

  5. State whether each statement is True or False. For any that are False, write the correct answer. Understanding

    1. 34 × 32 = 38
    2. 56 ÷ 52 = 53
    3. 23 × 43 = 86
    4. 75 ÷ 75 = 0
    5. 64 × 60 = 64
    6. 103 × 102 = 1005

  6. Multi-step index law problems. Problem Solving

    1. Simplify: (24 × 23) ÷ (25 ÷ 22)
    2. A spreadsheet formula multiplies values stored as powers of 3. Cell A1 contains 35 and Cell A2 contains 34. Write the result of A1 × A2 in index notation, then evaluate the answer.
    3. Jake says that 23 × 34 = 67. Is he correct? Explain your reasoning with full working.
    4. A rectangular paddock has an area of 59 m² and a length of 54 m. Write an expression for its width using index notation. Simplify, then evaluate.

  7. Simplify each expression involving three or more index terms with the same base. Fluency

    1. 32 × 33 × 34
    2. 58 ÷ 53 ÷ 52
    3. 21 × 22 × 23 × 24
    4. 710 ÷ 74 ÷ 72
    5. 43 × 40 × 45
    6. 69 ÷ 63 ÷ 60
    7. 92 × 93 × 91
    8. 1012 ÷ 104 ÷ 105

  8. Write an expression in index notation for each description, then simplify. Understanding

    1. The product of 43 and 4 to the power of six.
    2. 7 to the power of nine, divided by 7 to the power of four.
    3. The product of 25 and 23, divided by 26.
    4. Three to the power of eight, divided by the product of 32 and 33.

  9. Simplify both expressions, then use <, >, or = to compare them. Understanding

    1. 25 × 23   ___   24 × 24
    2. 39 ÷ 33   ___   37 ÷ 31
    3. 54 × 52   ___   53 × 53
    4. 47 ÷ 43   ___   46 ÷ 42

  10. Apply index laws to real-world situations. Problem Solving

    1. A scientist cultures bacteria. On Monday she has 24 bacteria. By Wednesday there are 24 × 23 bacteria. By Friday the amount triples again to 24 × 23 × 21. Write the total on Friday as a single power of 2, then evaluate.
    2. A storage warehouse fills 38 containers. A truck removes 33 containers each trip. How many full trips can be made before the warehouse is empty? Write your answer using the division index law, then evaluate.
    3. Two students simplify 65 × 63 ÷ 64:
      • Aiden gets 64
      • Brianna gets 612
      Who is correct? Show full working.
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