Sometimes tested Papers 1 & 2

Number: Indices, Surds, Standard Form, Bounds, Primes and HCF/LCM

Index laws, surds and standard form let you rewrite numbers into exact, comparable forms, while prime factorisation, bounds and estimation give fast no-calculator shortcuts for eliminating multiple-choice options. Fluency here underpins almost every TMUA question.

Part of the TMUA syllabus — revision for the Test of Mathematics for University Admission (TMUA), the UAT-UK maths test used by Cambridge, Oxford, Imperial, UCL, LSE, Warwick and Durham. No calculator; multiple choice.

Key points

  • The index laws are the engine of this topic: multiply by adding indices, divide by subtracting, and raise a power to a power by multiplying (a^m × an = am+n, (a^m)n = amn), with a0 = 1 for any a ≠ 0.
  • Negative and fractional indices turn reciprocals and roots into powers, so every surd becomes an index you can manipulate (a-n = 1/an, a1/2 = √(a), a3/2 = (√(a))3).
  • Simplify a surd by pulling out the largest square factor (√(50) = √(25 × 2) = 5 × √(2)); a surd is fully simplified when no square factor remains under the root.
  • Rationalise a denominator containing a surd by multiplying top and bottom by its conjugate, using (a + √(b))(a - √(b)) = a2 - b to clear the root.
  • Standard form writes a number as a × 10n with 1 ≤ a < 10; a positive n gives a large number and a negative n a small one.
    4.2 × 10-3 = 0.0042
  • Every integer above 1 has a unique prime factorisation; read the HCF as the lowest power of each shared prime and the LCM as the highest power of every prime that appears.
  • For two positive integers, HCF × LCM = product of the numbers, so the two quantities and their product determine each other instantly.
  • Bounds capture rounding error: a value given as 6 to the nearest whole number lies in 5.5 ≤ value < 6.5, and you combine bounds carefully to bound a sum, product or quotient.

Formulae

a^m × an = am+n

Multiplying two powers of the same base: add the indices.

a^m / an = am-n

Dividing two powers of the same base: subtract the indices.

am/n = (a1/n)^m

Turning a fractional index into an nth root raised to a power (a1/2 = √(a)).

√(a) × √(b) = √(ab)

Combining or simplifying surds; likewise √(a)/√(b) = √(a/b).

HCF(a,b) × LCM(a,b) = a × b

Finding the HCF or LCM of two positive integers when the other and the product are known.

1/(a + √(b)) = (a - √(b))/(a2 - b)

Rationalising a denominator by multiplying by the conjugate.

Definitions

Surd
A root that is irrational and kept in exact form, such as √(2) or √(3); simplifying it means removing square factors, not finding a decimal approximation.
Standard form
A number written as a × 10n where 1 ≤ a < 10 and n is an integer, used to handle very large or very small numbers without a calculator.
Upper and lower bound
The largest and smallest values a rounded quantity could actually take; a measurement x given to the nearest unit u satisfies x - u/2 ≤ value < x + u/2.
HCF and LCM
The highest common factor is the largest integer that divides two numbers exactly; the lowest common multiple is the smallest positive integer that both numbers divide into.

Worked examples

1

Write (√(3))5 / 9 in the form 3^k, where k is a fraction, and state the value of k.

  1. 1

    Write the root as a power:

    (√(3))5 = 35/2
  2. 2

    Rewrite the denominator:

    9 = 32
  3. 3

    Dividing subtracts indices:

    35/2/32 = 35/2-2
  4. 4

    Simplify the index:

    5/2 - 2 = 1/2
  5. 5

    Match the required form:

    3^k = 31/2

Answer: k = 1/2 (the expression equals 31/2 = √(3)).

2

Two positive integers have product 34 × 52 × 7 and highest common factor 3 × 5. Find their lowest common multiple, giving your answer as an integer.

  1. 1

    Use the identity:

    HCF × LCM = product
  2. 2

    Rearrange for LCM:

    LCM = product / HCF
  3. 3

    Substitute values:

    LCM = (34 × 52 × 7)/(3 × 5)
  4. 4

    Subtract indices:

    LCM = 33 × 5 × 7
  5. 5

    Evaluate:

    LCM = 27 × 5 × 7

Answer: LCM = 945.

Common mistakes

  • ×Mixing up the index laws: a^m × an adds indices but (a^m)n multiplies them, so 23 × 24 = 27 while (23)4 = 212.
  • ×Treating √(a + b) as √(a) + √(b); roots do not distribute over addition, since √(9 + 16) = 5, not 3 + 4 = 7.
  • ×Reading a negative index as a negative number: a-n = 1/an makes the value smaller, so 2-3 = 1/8, not -8.
  • ×Combining bounds the obvious way: the lower bound of a - b uses the lower bound of a but the UPPER bound of b (and similarly for division).
  • ×Standard form slips: writing 0.0042 as 4.2 × 103 instead of 4.2 × 10-3, or leaving a outside the range 1 ≤ a < 10.

No-calculator tips

  • ✓Convert every root and reciprocal to an index first; index laws turn a messy surd expression into simple addition and subtraction of fractions.
  • ✓Prime-factorise once, then read the HCF, the LCM, whether a number is a perfect square (all indices even) and any simplified surd straight off the factor tree.
  • ✓For estimation questions round each quantity to 1 significant figure, then track the power of 10 on its own; that alone eliminates most of the options.
  • ✓Rationalise with the conjugate so the denominator becomes an integer via (a + √(b))(a - √(b)) = a2 - b, making the answer options directly comparable.
  • ✓Sanity-check size and sign: a fractional index between 0 and 1 gives a value between 1 and the base, and a negative index gives a value below 1, so you can bin the choices before any full calculation.

Test yourself

Original practice questions, no calculator. Work each out before revealing the answer.

Q1.Which of the following is the value of (6 × 105) × (5 × 10-8) written correctly in standard form?

  • A. 30 × 10-3
  • B. 3 × 10-3
  • C. 3 × 10-2
  • D. 3 × 10-4
Show answer

Answer: C — 3 × 10-2

Multiply the numbers (6*5 = 30) and add the powers (105 × 10-8 = 10-3), giving 30 × 10-3 = 3 × 10-2. The value 30 × 10-3 is numerically equal but is not standard form, since 30 is not between 1 and 10.

Q2.A rectangle has length 12 cm and width 8 cm, each measured to the nearest centimetre. What is the upper bound of its area, in cm2?

  • A. 106.25
  • B. 96
  • C. 86.25
  • D. 117
Show answer

Answer: A — 106.25

To the nearest cm the bounds are ±0.5, so the upper bound of the area is 12.5 × 8.5 = 106.25. Using 11.5 × 7.5 = 86.25 mistakenly takes the lower bounds, and 13 × 9 = 117 wrongly uses a whole unit instead of 0.5.

Q3.Evaluate 16^(3/4) - 8^(2/3).

  • A. 12
  • B. 2
  • C. 32
  • D. 4
Show answer

Answer: D — 4

16^(3/4) = (16^(1/4))3 = 23 = 8 and 8^(2/3) = (8^(1/3))2 = 22 = 4, so the difference is 8 - 4 = 4. You cannot combine 23 - 22 into 2^(3-2) = 2, because the index law for subtracting powers applies to division, not subtraction.

Read this topic in the official UAT-UK TMUA content specification →

Keep going

All TMUA topics