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Mathematics 4024

Quick revision notes · Mathematics 4024 · Chapter 2 of 35

Indices

Powers and roots: the four laws, zero/negative/fractional powers, algebra with powers, and index equations.

4 sectionsWorked examples Common mistakes
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At a glance

Powers and roots: the four laws, zero/negative/fractional powers, algebra with powers, and index equations.

  1. The laws of indices
  2. Zero, negative and fractional powers
  3. Indices in algebra
  4. Equations with the unknown in the power

1The laws of indices

  • The index (power) says how many times a number is multiplied by itself: 25 = 2 × 2 × 2 × 2 × 2 = 32.
  • The laws only work when the base is the same: 23 × 24 = 27, but 23 × 34 cannot be combined.
  • Numbers in front are dealt with normally: 3x2 × 4x5 = 12x7.

Formulas

am × an = am+nmultiply → add the powers
am ÷ an = am−ndivide → subtract the powers
(am)n = amnpower of a power → multiply
(ab)n = an bneverything inside is raised

Worked example

Simplify 25 × 23 ÷ 24 and find its value.

  1. Add: 5 + 3 = 8.
  2. Subtract: 8 − 4 = 4.
  3. 24 = 16.

24 = 16

2Zero, negative and fractional powers

  • a0 = 1 for any a except 0: 70 = 1, (−5)0 = 1.
  • A negative power means 'one over': a−n = 1/an. It never makes the answer negative: 2−3 = 1/8.
  • A fraction raised to a negative power flips: (2/3)−2 = (3/2)2 = 9/4.
  • A fractional power is a root: a1/2 = √a, a1/3 = ∛a, and am/n = (ⁿ√a)m — do the root first.
  • Negative fractional: do all three — flip, root, power. 8−2/3 = 1/(∛8)2 = 1/4.

For a−m/n

  1. Negative sign? Flip the number (write 1 over it).
  2. Bottom of the fraction? Take that root.
  3. Top of the fraction? Raise to that power.

Formulas

a0 = 1
a−n = 1/anone over
a1/n = ⁿ√aroot
am/n = (ⁿ√a)mroot first, then power

Worked example

Find the value of (27/8)−2/3.

  1. Flip: (8/27)2/3.
  2. Cube root: ∛8 = 2 and ∛27 = 3, giving 2/3.
  3. Square: (2/3)2.

4/9

Exam tip: Know the powers of 2 (2, 4, 8, 16, 32, 64, 128), 3 (3, 9, 27, 81, 243) and 5 (5, 25, 125, 625) by heart.

3Indices in algebra

  • Deal with the numbers and each letter separately.
  • (3x2)3 = 33 × x6 = 27x6 — the number in front is cubed too.
  • 12a5 ÷ 3a−2 = 4a5−(−2) = 4a7.
  • (2x3/5)2 = 4x6/25: square the top and the bottom.

Worked example

Simplify (2x3 y)4 ÷ 8x5 y2.

  1. Expand the bracket: 16x12 y4.
  2. Divide the numbers: 16 ÷ 8 = 2.
  3. Subtract powers: x12−5 = x7, y4−2 = y2.

2x7 y2

4Equations with the unknown in the power

  • Write both sides as powers of the same base, then set the powers equal.
  • Useful bases: 4 = 22, 8 = 23, 16 = 24, 32 = 25; 9 = 32, 27 = 33, 81 = 34; 25 = 52, 125 = 53.
  • 1/8 = 2−3 and √2 = 21/2 — negative and fractional powers often appear here.
  • No logarithms are needed (or expected) at this level.

How to do it

  1. Choose a common base (usually 2, 3 or 5).
  2. Rewrite each side as basepower, using (am)n = amn.
  3. Set the powers equal and solve the linear equation.

Worked example

Solve 4x+1 = 8x.

  1. Base 2: (22)x+1 = (23)x.
  2. 22x+2 = 23x.
  3. 2x + 2 = 3x.

x = 2

Exam tip: If one side is a fraction like 1/27, write it as a negative power first: 3−3.

✗Common mistakes

Don’t writeWrite instead
23 × 24 = 212 Add the powers: 27
23 × 24 = 47 The base stays 2: 27
4−2 = −16 4−2 = 1/16
(3x2)3 = 3x6 Cube the 3 too: 27x6
50 = 0 50 = 1
82/3 = (82)1/3 worked out the long way Root first is easier: (∛8)2 = 4

★In the exam

  • Non-calculator questions on indices are common — practise without a calculator.
  • For 'find n' questions, write every number as a power of the same base before comparing.
  • Check by substituting your answer back in.