← IGCSE / O Level Mathematics

IGCSE / O Level Mathematics · Topic 1 Number · Chapter 7 · pp. 62–72

Indices and standard form

An index is shorthand for repeated multiplication: 53 means 5 × 5 × 5. Three short laws let you multiply, divide and raise powers without writing anything out. Standard form uses powers of 10 to write the distance to the Sun or the width of an atom in a few characters.

Learning objectives

  1. Use the three laws of indices with positive indices.
  2. Understand and use zero and negative indices.
  3. Solve simple exponential equations such as 2x = 32.
  4. Write very large and very small numbers in standard form, and calculate with them.
  5. Understand and use fractional indices. (Extended)

Book exercises: 7.1 to 7.8 and Student assessments 1 to 3 (pp. 62–72).

The three laws

LawRuleExample
1. Multiplyingam × an = am+n (add)43 × 42 = 45
2. Dividingam ÷ an = am−n (subtract)25 ÷ 23 = 22
3. Power of a power(am)n = amn (multiply)(42)3 = 46

💡 The laws only work when the base is the same. 23 × 32 can't be combined.

444 × 44 = 4⁵ (five 4s) 4³: three 4s 4²: two 4s 3 + 2 = 5
Why the first law adds: three 4s times two 4s is five 4s multiplied together.

📘 Example 1 · Evaluate 33 × 34

  1. Same base, multiplying, so add the indices: 33+4 = 37.
  2. Evaluate: 37 = 2187.

📘 Example 2 · Simplify

  1. Top: 56+7 = 513.
  2. Bottom: 52+8 = 510.
  3. Divide by subtracting: 513−10 = 53 (= 125).

✏️ Try it

  1. Simplify 52 × 54 × 53.
  2. Simplify (102)5.
  3. Write 3 × 3 × 4 × 4 × 4 × 5 × 5 using indices.
Answers

1. 59 · 2. 1010 · 3. 32 × 43 × 52

Next: zero and negative indices →

Zero and negative indices

🔑 Key ideas

  • a0 = 1 for any non-zero a, because am ÷ am is both 1 and a0.
  • a−m = : a negative index means “one over”.
  • A negative index does not make the number negative: 2−3 = , not −8.
2³ = 8 ÷ 2 2² = 4 ÷ 2 2¹ = 2 ÷ 2 2⁰ = 1 ÷ 2 2⁻¹ = ½ ÷ 2 2⁻² = ¼
Each step down the ladder divides by 2. Keep going past 2¹ and you reach 2⁰ = 1, then fractions.

📘 Example 3 · Evaluate 16 × 2−3

  1. Rewrite the negative index: 2−3 = = .
  2. Multiply: 16 × = 2.

📘 Example 4 · Evaluate

  1. Same base, dividing, so subtract: 7−3 − (−4) = 7−3+4.
  2. = 71 = 7.

✏️ Try it

  1. 52 × 5−2
  2. 3−2
  3. 5 × 10−3
Answers

1. 50 = 1 · 2. · 3. 0.005

Next: exponential equations →

Exponential equations

An exponential equation has the unknown in the index, such as 2x = 32.

1. Same base32 = 2⁵ 2. Match indices2ˣ = 2⁵ → x = 5 3. Solve & check2⁵ = 32 ✔
Write both sides as a power of the same base. When the bases match, the indices must be equal.

📘 Example 5 · Solve 3m − 1 = 81

  1. Write 81 as a power of 3: 81 = 34.
  2. So 3m − 1 = 34, and the indices are equal: m − 1 = 4.
  3. Solve: m = 5.
  4. Check: 35−1 = 34 = 81. ✔

📘 Example 6 · Solve 2−x =

  1. 256 = 28, so = 2−8.
  2. 2−x = 2−8 gives −x = −8.
  3. x = 8.

✏️ Try it

  1. 4x = 64
  2. 3z + 2 = 27
  3. 10z + 1 = 1
Answers

1. 64 = 43, x = 3 · 2. z + 2 = 3, z = 1 · 3. 1 = 100, so z + 1 = 0, z = −1

Next: standard form →

Standard form

🔑 Key ideas

  • Standard form is A × 10n with 1 ≤ A < 10 and n a whole number.
  • Big numbers (10 or more) have a positive n. Numbers between 0 and 1 have a negative n.
  • n counts how many places the decimal point moves to make A.
  • 31 × 102 and 0.31 × 104 equal 3100 but are not in standard form. Only 3.1 × 103 is.
72000 1234 = 7.2 × 10⁴ 4 places left → n = 4 00032 123 = 3.2 × 10⁻³ 3 places right → n = −3 Plain dot: where the point starts · coloured dot: where it ends
Big numbers: the point moves left and n is positive. Small numbers: the point moves right and n is negative.

📘 Example 7 · Write 72 000 in standard form

  1. Put the point after the first non-zero digit: A = 7.2.
  2. Count the places from 7.2 back to 72 000: 4 places, and the number is big, so n = 4.
  3. 72 000 = 7.2 × 104.

📘 Example 8 · Write 1.8 × 10−4 as an ordinary number

  1. A negative index means divide: 1.8 ÷ 104 = 1.8 ÷ 10 000.
  2. Move the point 4 places left: 0.000 18.

🧪 Try it: standard form converter

    📘 Example 9 · Order from largest: 3.6 × 10−3, 5.2 × 10−5, 1 × 10−2, 8.35 × 10−2

    1. Compare the powers of 10 first: the biggest index is the biggest number (−2 > −3 > −5).
    2. Two numbers share 10−2, so compare their A: 8.35 > 1.
    3. Order: 8.35 × 10−2, 1 × 10−2, 3.6 × 10−3, 5.2 × 10−5.

    ✏️ Try it

    1. Write 784 000 000 000 in standard form.
    2. Write 0.000 053 in standard form.
    3. Is 82.3 × 106 in standard form? If not, fix it.
    Answers

    1. 7.84 × 1011 · 2. 5.3 × 10−5 · 3. No, A must be below 10: 8.23 × 107

    Next: calculating in standard form →

    Calculating in standard form

    OperationWhat to do
    × multiplyMultiply the A parts, add the indices, then tidy A back into 1 ≤ A < 10.
    ÷ divideDivide the A parts, subtract the indices, then tidy.
    + − add or subtractFirst rewrite both with the same power of 10, then add or subtract the A parts, then tidy.

    📘 Example 10 · (2.4 × 104) × (5 × 107)

    1. Multiply the A parts: 2.4 × 5 = 12.
    2. Add the indices: 104+7 = 1011. So far: 12 × 1011.
    3. Tidy: 12 = 1.2 × 101, so the answer is 1.2 × 1012.

    📘 Example 11 · (6.4 × 107) ÷ (1.6 × 103)

    1. Divide the A parts: 6.4 ÷ 1.6 = 4.
    2. Subtract the indices: 107−3 = 104.
    3. Answer: 4 × 104.

    📘 Example 12 · (3.8 × 106) + (8.7 × 104)

    1. Same power of 10: 3.8 × 106 = 380 × 104.
    2. Add the A parts: 380 + 8.7 = 388.7, so 388.7 × 104.
    3. Tidy: 3.887 × 106.

    ✏️ Try it

    1. (4.4 × 103) × (2 × 105)
    2. (3.8 × 108) ÷ (1.9 × 106)
    3. Light travels at 3 × 108 m/s and takes 8 minutes to reach Earth. How far is the Sun (3 s.f.)?
    Answers

    1. 8.8 × 108 · 2. 2 × 102 · 3. 480 s × 3 × 108 = 1.44 × 1011 m

    Next: fractional indices →

    Fractional indices Extended

    🔑 Key ideas

    • a1/2 = √a and a1/3 = ∛a. In general a1/n is the nth root.
    • am/n: the bottom of the fraction is the root, the top is the power.
    • Take the root first: the numbers stay small. 253/2 = (√25)3 = 53.
    25 32 Bottom 2 = root√25 = 5 Top 3 = power5³ = 125 25^(3/2) = (√25)³ = 5³ = 125
    Read a fractional index bottom first: root, then power.

    📘 Example 13 · Evaluate 161/4

    1. The bottom of the index is 4, so take the fourth root.
    2. Which number to the power 4 is 16? 2 × 2 × 2 × 2 = 16, so 161/4 = 2.

    📘 Example 14 · Solve 125x = 5

    1. 125 = 53, so 5 = ∛125 = 1251/3.
    2. 125x = 1251/3, so x = .

    📘 Example 15 · Evaluate 27−2/3

    1. Negative index: 27−2/3 = .
    2. Root first: ∛27 = 3.
    3. Then power: 32 = 9, so the answer is .

    ✏️ Try it

    1. 271/3
    2. 43/2
    3. Solve 16x = 4.
    Answers

    1. 3 · 2. (√4)3 = 8 · 3. x =

    Next: summary and quiz →

    Chapter summary

    • am × an = am+n · am ÷ an = am−n · (am)n = amn. Same base only.
    • a0 = 1 and a−m = 1 ÷ am.
    • Exponential equations: write both sides with the same base, then equate the indices.
    • Standard form A × 10n, 1 ≤ A < 10. Small numbers have negative n.
    • × add indices, ÷ subtract indices, + and − need the same power of 10 first. Tidy A at the end.
    • am/n: bottom is the root, top is the power.

    📄 Practice questions

    The book's Exercises 7.1 to 7.8 and Student assessments 1 to 3 are in the printable assignment for this chapter.

    Mini-quiz

    1. Simplify 4³ × 4².
    2. Simplify (4²)³.
    3. What is the value of 7⁰?
    4. What is the value of 2⁻³?
    5. Solve 3^(m − 1) = 81.
    6. What is 72 000 in standard form?
    7. What is (2.4 × 10⁴) × (5 × 10⁷) in standard form?
    8. What is 25^(3/2)?

    Source: Cambridge IGCSE Mathematics Core and Extended (4th edition, Pimentel and Wall), Chapter 7 (Indices and standard form). 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.