← IGCSE / O Level Mathematics

IGCSE / O Level Mathematics · Topic 2 Algebra and graphs · Chapter 12 · pp. 122–125

Algebraic indices

Chapter 7 used the laws of indices with numbers such as 53. Here the same three laws work with letters: d3 × d4 = d7, whatever d stands for. Zero, negative and fractional indices follow from the same laws.

Learning objectives

  1. Use the three laws of indices to simplify algebraic terms.
  2. Simplify terms that have numbers (coefficients) and several letters.
  3. Use zero and negative indices with letters.
  4. Change between fractional indices and roots, and simplify with them. (Extended)

Book exercises: 12.1 to 12.3 and Student assessments 1 and 2 (pp. 122–125).

The three laws with letters

LawRuleExample
1. Multiplyingam × an = am+n (add)d3 × d4 = d7
2. Dividingam ÷ an = = am−n (subtract)m4 ÷ m2 = m2
3. Power of a power(am)n = amn (multiply)(p2)4 = p8

💡 The laws only join powers of the same letter. x3 × y2 stays as x3y2.

ddd × dddd = d⁷ d³: three d's d⁴: four d's 3 + 4 = 7 d's
Why law 1 adds: three d's times four d's is seven d's multiplied together.
Powers of the same letter? Then ask: what is the sign? × Multiplying aᵐ × aⁿ = aᵐ⁺ⁿ add the indices ÷ Dividing aᵐ ÷ aⁿ = aᵐ⁻ⁿ subtract the indices ( )ⁿ Bracket power (aᵐ)ⁿ = aᵐⁿ multiply the indices Numbers in front (coefficients) are worked out as normal: 4a² × 3a³: 4 × 3 = 12, and a²⁺³ = a⁵, so 12a⁵.
Which law? Look at the operation between the powers of the same letter.

📘 Example 1 · Simplify d3 × d4 (book)

  1. Same letter, multiplying: law 1, add the indices.
  2. d3 + 4 = d7.

📘 Example 2 · Simplify (book)

  1. Top, law 3: (p2)4 = p2×4 = p8.
  2. Bottom, law 1: p2 × p4 = p2+4 = p6.
  3. Divide, law 2: p8−6 = p2.

📘 Example 3 · Simplify

  1. Numbers: 10 ÷ 2 = 5.
  2. x: x5−2 = x3.
  3. y: y on its own is y1, so y3−1 = y2.
  4. Put it together: 5x3y2.
  5. Check with x = 1, y = 2: 10 × 8 ÷ (2 × 2) = 20, and 5 × 1 × 4 = 20. ✔

📘 Example 4 · Simplify (3m2n)3

  1. The power 3 applies to every part inside the bracket: 33 × (m2)3 × n3.
  2. 33 = 27.
  3. Law 3: (m2)3 = m6.
  4. Answer: 27m6n3.

⚠️ Common slip: (3m2)3 is 27m6, not 3m6 or 9m6. The number is cubed too.

🧪 Try it: laws of indices with letters

Indices can be whole numbers, negative (−2) or fractions (1/2, −3/4).

    ✏️ Try it

    1. Simplify y4 × y6.
    2. Simplify 15p7 ÷ 5p3.
    3. Simplify (2a3b)2 × a.
    Answers

    1. y10 · 2. 3p4 · 3. 4a6b2 × a1 = 4a7b2

    Next: zero and negative indices →

    Zero and negative indices

    🔑 Key ideas

    • a0 = 1 (a ≠ 0). Any term to the power zero is 1.
    • a−m = : a negative index means “one over”.
    • The index belongs only to the letter it touches: 6x−2 = , and 5b0 = 5.
    Proofa0 = 1a−m = 1 ÷ am
    Start = am−m = a0 (law 2)a−m = a0−m
    But also = 1 (anything ÷ itself)a0−m = (law 2 backwards)
    Soa0 = 1= , so a−m =
    x³ = x × x × x ÷ x x² = x × x ÷ x x¹ = x ÷ x x⁰ = 1 ÷ x x⁻¹ = 1 ÷ x ÷ x x⁻² = 1 ÷ x²
    Each step down divides by x and takes 1 off the index. Past x¹ you reach x⁰ = 1, then “one over”.

    📘 Example 5 · Simplify (k3)2 × (k−2)4

    1. Law 3 on each bracket: (k3)2 = k6 and (k−2)4 = k−8.
    2. Law 1: k6 + (−8) = k−2.
    3. Without a negative index: k−2 = .

    📘 Example 6 · Simplify (w4 ÷ w−2)2

    1. Inside the bracket, law 2: w4 − (−2) = w4+2 = w6.
    2. Law 3: (w6)2 = w12.

    📘 Example 7 · Simplify

    1. y0 = 1, so the top is y5.
    2. Law 2: y5 − (−1) = y6.
    3. Answer: y6.

    💡 Minus minus: dividing by a negative power is the same as multiplying by the positive power. = a3.

    ✏️ Try it

    1. Simplify n5 × n−5.
    2. Simplify (a−1)4 and write it without a negative index.
    3. Simplify 8b−3 ÷ 2b2.
    Answers

    1. n0 = 1 · 2. a−4 = · 3. 4b−5 =

    Next: fractional indices →

    Fractional indices Extended

    🔑 Key ideas

    • a1/n = n√a. So a1/2 = √a and a1/3 = ∛a.
    • am/n = n√(am) or (n√a)m: the bottom is the root, the top is the power.
    • Why: am/n = (am)1/n = n√(am), and also am/n = (a1/n)m = (n√a)m. Both use law 3.
    • All three laws still work: add, subtract or multiply the fractions.
    a m n Top m = power raise to the power m Bottom n = root take the nth root ⁿ√(aᵐ) or (ⁿ√a)ᵐ Example: b with index 2/5 is (⁵√b)², and (∛a)⁴ is a with index 4/3
    Read a fractional index as a fraction: the top is the power, the bottom is the root.

    📘 Example 8 · Express (∛a)4 in the form am/n (book)

    1. Cube root is the power : ∛a = a1/3.
    2. So (∛a)4 = (a1/3)4.
    3. Law 3: × 4 = , so the answer is a4/3.

    📘 Example 9 · Express b2/5 in the form (n√b)m (book)

    1. Split the index: b2/5 = (b1/5)2.
    2. b1/5 is the fifth root: 5√b.
    3. Answer: (5√b)2.

    📘 Example 10 · Simplify (book)

    1. Top, law 1: + = + = , so p5/6.
    2. The bottom p is p1.
    3. Law 2: − 1 = −.
    4. Answer: p−1/6, which can also be written (6√p)−1 or .

    📘 Example 11 · Simplify (8x6)1/3

    1. The power applies to both parts: 81/3 × (x6)1/3.
    2. 81/3 = ∛8 = 2.
    3. Law 3: 6 × = 2, so x2.
    4. Answer: 2x2. Check: (2x2)3 = 8x6. ✔

    ✏️ Try it

    1. Write (5√a)2 in the form am/n.
    2. Write c3/4 in the form (n√c)m.
    3. Simplify x2/3 ÷ x1/6.
    Answers

    1. a2/5 · 2. (4√c)3 · 3. − = = , so x1/2 = √x

    Next: summary and quiz →

    Chapter summary

    • am × an = am+n · am ÷ an = am−n · (am)n = amn. Same letter only.
    • Numbers in front are multiplied or divided as normal. A bracket power applies to every part inside.
    • a0 = 1 and a−m = . The index belongs only to the letter it touches.
    • a1/n = n√a and am/n = (n√a)m: bottom is the root, top is the power. (Extended)

    📄 Practice questions

    The book's Exercises 12.1 to 12.3 and Student assessments 1 and 2 are in the printable assignment for this chapter.

    Mini-quiz

    1. Simplify x³ × x⁵.
    2. Simplify m⁹ ÷ m³.
    3. Simplify (p⁴)³.
    4. Simplify 3a² × 4a⁵.
    5. Simplify (2y³)².
    6. Simplify 10x⁶y⁴ ÷ 5x²y.
    7. Simplify 7b⁰.
    8. Write n⁻³ without a negative index.
    9. Simplify a⁻² × a⁵.
    10. Which is the cube root of x, ∛x, written as a power of x?
    11. Write (⁴√y)³ in the form y^(m/n).
    12. Simplify a^(1/2) × a^(1/3).

    Source: Cambridge IGCSE Mathematics Core and Extended (4th edition, Pimentel and Wall), Chapter 12 (Algebraic indices). Explanations, the Try it answers, the tool and the quiz are Guru Jazzy's own. This page is not made or endorsed by Cambridge or Hodder Education.