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O-Level Mathematics · Secondary 3 and 4

Indices and Standard Form

Indices give a short way to write repeated multiplication, such as 2⁵ for 2 × 2 × 2 × 2 × 2. In O-Level Mathematics you apply the laws of indices to numbers and algebraic terms, work with zero, negative and fractional indices, solve simple equations where the unknown is in the index, and write very large and very small numbers in standard form. These skills are used in exponential graphs, in science calculations and in Additional Mathematics, where indices lead on to logarithms.

What the syllabus expects

  • Use the laws of indices for multiplication, division and powers of powers
  • Use zero, negative and fractional indices, including a⁰ = 1, a⁻ⁿ = 1/aⁿ and a^(1/n) as the nth root of a
  • Simplify algebraic expressions involving indices, giving answers with positive indices
  • Solve simple equations with the unknown in the index by writing both sides as powers of the same base
  • Write numbers in standard form A × 10ⁿ, where 1 ≤ A < 10 and n is an integer, and calculate with them

The laws of indices

For any non-zero base a, the main laws are: aᵐ × aⁿ = a^(m + n), aᵐ ÷ aⁿ = a^(m − n), (aᵐ)ⁿ = a^(mn), and (ab)ⁿ = aⁿbⁿ. The first two laws only apply when the bases are the same, so 2³ × 2⁴ = 2⁷ = 128, but 2³ × 3⁴ cannot be combined into a single power. With algebraic terms, deal with the numbers and each letter separately: 3x²y × 4x⁵y³ = 12x⁷y⁴, because 3 × 4 = 12 and the indices of each letter are added. When a bracket is raised to a power, every factor inside is raised to that power: (2x³)⁴ = 2⁴x¹² = 16x¹². A common error is to multiply the 2 by 4 instead of raising it to the power 4. Division works the same way: 12x⁷ ÷ 4x² = 3x⁵.

Zero, negative and fractional indices

The laws explain the meaning of other indices. Since a³ ÷ a³ = 1 and also a³ ÷ a³ = a⁰, we have a⁰ = 1 for any non-zero a. Similarly a⁻ⁿ = 1/aⁿ, so 5⁻² = 1/25 and 2x⁻³ = 2/x³. Notice that only x is raised to −3 in 2x⁻³, so the 2 stays in the numerator. A fractional index means a root: a^(1/n) is the nth root of a, so 16^(1/2) = 4 and 27^(1/3) = 3. For a^(m/n), take the root first and then the power, which keeps the numbers small: 27^(2/3) = (∛27)² = 3² = 9 and 32^(3/5) = 2³ = 8. Combine these ideas for harder values: 25^(−1/2) = 1/25^(1/2) = 1/5, and (8/27)^(−2/3) = (27/8)^(2/3) = (3/2)² = 9/4.

Equations with the unknown in the index

To solve an equation such as 2ˣ = 32, write both sides as powers of the same base and then equate the indices: 32 = 2⁵, so x = 5. The same idea works with negative and fractional answers. For 9ˣ = 1/3, write 9 = 3² and 1/3 = 3⁻¹, so 3^(2x) = 3⁻¹, giving 2x = −1 and x = −1/2. For 4^(x + 1) = 8, write 4 = 2² and 8 = 2³, so 2^(2x + 2) = 2³, giving 2x + 2 = 3 and x = 1/2. Choose the smallest base that works for both sides, usually 2, 3 or 5. Equating indices is only valid once both sides are single powers of the same base, so simplify products and quotients first using the laws of indices.

Standard form

A number in standard form is written as A × 10ⁿ, where 1 ≤ A < 10 and n is an integer. So 4 560 000 = 4.56 × 10⁶ and 0.000 304 = 3.04 × 10⁻⁴. For small numbers, n is negative and counts how many places the decimal point moves to the right to reach a number between 1 and 10. To multiply or divide, handle the numbers and the powers of 10 separately, then adjust: (4.8 × 10⁵) × (6 × 10⁻³) = 28.8 × 10² = 2.88 × 10³, and (4.8 × 10⁵) ÷ (6 × 10⁻³) = 0.8 × 10⁸ = 8 × 10⁷. To add or subtract, first rewrite the numbers with the same power of 10: 3.2 × 10⁴ + 5 × 10³ = 3.2 × 10⁴ + 0.5 × 10⁴ = 3.7 × 10⁴. Your calculator shows standard form in its own style, but you must write the answer in the A × 10ⁿ form.

Worked example

Simplify (2x³y⁻²)³ ÷ (4x⁵y⁻⁴), giving your answer with positive indices.

  1. Raise each factor in the first bracket to the power 3: 2³ = 8, (x³)³ = x⁹ and (y⁻²)³ = y⁻⁶, so the first part is 8x⁹y⁻⁶.
  2. Divide the numbers: 8 ÷ 4 = 2.
  3. Subtract the indices of x: x⁹ ÷ x⁵ = x⁴.
  4. Subtract the indices of y: y⁻⁶ ÷ y⁻⁴ = y^(−6 − (−4)) = y⁻².
  5. Combine to get 2x⁴y⁻², then use y⁻² = 1/y² to remove the negative index.

Answer: 2x⁴/y²

Common mistakes

  • Multiplying the bases. 2³ × 2⁴ is 2⁷, not 4⁷. The base stays the same and only the indices are added.
  • Adding indices for a power of a power. (x³)² = x⁶, because it means x³ × x³. Writing x⁵ confuses this law with the multiplication law.
  • Thinking a⁰ = 0. Any non-zero number to the power 0 is 1, so 7⁰ = 1 and (3x)⁰ = 1.
  • Applying a negative index to the coefficient as well. 2x⁻³ = 2/x³, but (2x)⁻³ = 1/(8x³). Look at whether there is a bracket.
  • Treating a negative index as a negative number. 16^(−1/2) = 1/4, not −4. A negative index means a reciprocal, not a change of sign.
  • Leaving A outside the range in standard form. 28.8 × 10² and 0.8 × 10⁸ are correct values but are not in standard form. Adjust them to 2.88 × 10³ and 8 × 10⁷.

Exam tips

  • Write every term as a power of a prime base before simplifying expressions such as 4ⁿ × 8 ÷ 2ⁿ. Once everything is a power of 2, the laws apply directly.
  • For fractional indices without a calculator, take the root before the power. 64^(2/3) is (∛64)² = 4² = 16, which is far easier than finding 64² first.
  • When a question says give your answer in standard form, check that the number in front is at least 1 and less than 10, and that you have written × 10ⁿ rather than copying a calculator display.

Practice questions

1. Without a calculator, evaluate 27^(2/3) and 16^(−1/2).

Show answer

27^(2/3) = (∛27)² = 3² = 9. 16^(−1/2) = 1/√16 = 1/4.

2. Solve 3^(2x + 1) = 1/27.

Show answer

x = −2. Write 1/27 as 3⁻³, so 2x + 1 = −3, which gives 2x = −4 and x = −2. Check: 3^(2(−2) + 1) = 3⁻³ = 1/27.

3. The mass of the Earth is about 5.97 × 10²⁴ kg and the mass of the Moon is about 7.35 × 10²² kg. Find, correct to 3 significant figures, (a) their total mass in standard form, (b) how many times heavier the Earth is than the Moon.

Show answer

(a) 6.04 × 10²⁴ kg. Write 7.35 × 10²² as 0.0735 × 10²⁴, then 5.97 + 0.0735 = 6.0435, so the total is 6.04 × 10²⁴ kg. (b) About 81.2 times. (5.97 ÷ 7.35) × 10^(24 − 22) = 0.81224 × 10² = 81.2.

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