Practice Maths

Power of a Power Law

Key Terms

Power of a power law
(am)n = am×n — when a power is raised to another power, multiply the indices.
Negative index
a−n = 1 ÷ an — a negative index gives the reciprocal; it does NOT make the number negative.
Hot Tip For power of a power, you multiply the indices (not add). Compare: am × an = am+n (add) vs (am)n = amn (multiply).

Worked Example

Question: Simplify (23)4 × 22 ÷ 26

Step 1 — Apply power of power law.
(23)4 = 212

Step 2 — Apply multiplication law.
212 × 22 = 214

Step 3 — Apply division law.
214 ÷ 26 = 28

Why the Power of a Power Law Multiplies Indices

The expression (32)4 means "raise 32 to the power of 4," which is 32 × 32 × 32 × 32. Using the multiplication law, this is 32+2+2+2 = 38. We added the index 2 four times, which is the same as multiplying: 2 × 4 = 8. So (am)n = am×n. This is one of the most powerful tools in index notation — it collapses doubly-nested powers into a single expression.

It's important not to confuse this with the multiplication law. Compare: am × an = am+n (add) versus (am)n = amn (multiply). The placement of the second index — whether it's next to the first as a multiplier, or written as a superscript on the bracket — determines which rule to use.

Key Tip: Power of a power = multiply the indices: (53)4 = 512. Multiplication of same-base terms = add the indices: 53 × 54 = 57. Don't mix these up!

Combining All Three Index Laws

When an expression uses all three laws, always deal with power-of-power first (it's inside brackets), then apply multiplication and division. Example: (23)2 × 25 ÷ 27. Step 1: (23)2 = 26. Step 2: 26 × 25 = 211. Step 3: 211 ÷ 27 = 24 = 16.

Sometimes you'll see two power-of-power terms divided: (34)2 ÷ (33)2 = 38 ÷ 36 = 32 = 9. Simplify each bracket first, then apply the division law.

Negative Indices: What Do They Mean?

Extending the division law: what happens when the subtracted index goes negative? Consider 23 ÷ 25 = 23−5 = 2−2. Writing it out: (2×2×2) ÷ (2×2×2×2×2) = 1/(2×2) = 1/4. So 2−2 = 1/4 = 1/22. The general rule: a−n = 1/an. A negative index means "reciprocal" — it does NOT make the number negative.

Common examples: 3−2 = 1/9; 5−1 = 1/5; 10−3 = 1/1000 = 0.001. Negative indices appear in scientific notation for very small numbers, like the size of an atom (around 10−10 metres).

Key Tip: a−n means 1/an, NOT −an. A negative index makes the number a fraction (between 0 and 1 for bases greater than 1), never negative. For example, 4−2 = 1/16, which is positive and less than 1.

Putting It All Together

The three index laws — multiplication (add), division (subtract), power of power (multiply) — form a complete toolkit. Real-world application: scientists write the distance to Andromeda as approximately 2.4 × 1022 metres. To square this distance for an area calculation: (2.4 × 1022)2 = 2.42 × (1022)2 = 5.76 × 1044 m2. Index laws make these colossal calculations manageable.

Mastery Practice

  1. Apply the power of a power law to simplify. Fluency

    1. (23)4
    2. (52)3
    3. (34)2
    4. (71)6
    5. (103)3
    6. (45)2
    7. (62)5
    8. (93)3

  2. Simplify using index laws. State which law you used each time. Fluency

    1. (32)3 × 34
    2. 56 ÷ (52)2
    3. (23)2 × 25 ÷ 27
    4. (43)2 ÷ 44
    5. (72)4 × 70
    6. 108 ÷ (102)3
    7. (63)3 × 62 ÷ 65
    8. (92)2 ÷ 91

  3. Simplify step by step using multiple laws. Write each step clearly. Fluency

    1. (23)2 × (22)3
    2. [(32)3 ÷ 34] × 32
    3. (54)2 ÷ [(53)2 × 5]
    4. (102)4 × 103 ÷ (105)2
    5. [(72)3 × 74] ÷ (75)2
    6. (43)2 ÷ (42)2 × 41
    7. (62)4 ÷ [(63)2 × 62]
    8. (93)2 × (92)3 ÷ 910

  4. Evaluate each expression involving negative indices. Write as a fraction in simplest form. Understanding

    1. 2−1
    2. 3−2
    3. 5−1
    4. 4−2
    5. 10−3
    6. 2−4
    7. 6−2
    8. 3−3

  5. For each simplification below, name the index law (or laws) used at each step. Understanding

    1. 53 × 54 = 57
    2. (24)3 = 212
    3. 79 ÷ 74 = 75
    4. (32)4 × 32 = 38 × 32 = 310
    5. 68 ÷ (62)3 = 68 ÷ 66 = 62
    6. (45)2 ÷ (43)3 = 410 ÷ 49 = 41 = 4

  6. Index laws in context. Problem Solving

    1. The distance from Earth to a nearby star is approximately (104)3 kilometres. Write this as a single power of 10, then as a standard number.
    2. A scientist writes the mass of a particle as 2−3 grams. Write this as a fraction and a decimal.
    3. Simplify fully, then evaluate: (23)2 ÷ (22)2 × 22
    4. Priya says (52)3 = 55. What mistake did she make? Give the correct answer.
    5. Write a single power of 2 equal to: (23)2 × 24 ÷ (25)2

  7. Apply the power of a power law, then evaluate the result. Fluency

    1. (22)3
    2. (31)4
    3. (102)2
    4. (23)3
    5. (52)2
    6. (41)5
    7. (25)1
    8. (32)2

  8. For each expression, first identify which law(s) apply, then simplify. Fluency

    1. 34 × 35
    2. (43)3
    3. 67 ÷ 64
    4. (25)2 × 23
    5. 59 ÷ (53)2
    6. 72 × (72)4
    7. (84)0
    8. (92)3 ÷ 95

  9. Find the missing value (represented by □) in each equation. Understanding

    1. (3□)4 = 38
    2. (53)□ = 515
    3. (2□)3 = 212
    4. (72)□ = 710
    5. (4□)5 = 420
    6. (63)□ = 69

  10. Multi-law challenge problems. Problem Solving

    1. A square garden has side length (32)2 metres. Write the area as a single power of 3, then evaluate in square metres.
    2. Two students simplify (43)2 ÷ 44:
      • Oscar gets 42
      • Nina gets 410
      Who is correct? Show full working to justify.
    3. Without evaluating, decide which is larger: (34)3 or (33)4. Explain your reasoning.
    4. Write an expression using the power of a power law that simplifies to 212. Write at least two different expressions.
See Answers ➔