Mathematics 4024 ยท Syllabus tracker
Indices
The laws of indices, zero, negative and fractional powers, simplifying algebra with powers and solving index equations.
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Indices on one sheet
For the night before. The full notes teach the topic; this is just what you must remember.
At a glance
Powers and roots: the four laws, zero/negative/fractional powers, algebra with powers, and index equations.
1The laws of indices
- The index (power) says how many times a number is multiplied by itself: 25 = 2 ร 2 ร 2 ร 2 ร 2 = 32.
- The laws only work when the base is the same: 23 ร 24 = 27, but 23 ร 34 cannot be combined.
- Numbers in front are dealt with normally: 3x2 ร 4x5 = 12x7.
Formulas
Worked example
Simplify 25 ร 23 รท 24 and find its value.
- Add: 5 + 3 = 8.
- Subtract: 8 โ 4 = 4.
- 24 = 16.
24 = 16
2Zero, negative and fractional powers
- a0 = 1 for any a except 0: 70 = 1, (โ5)0 = 1.
- A negative power means 'one over': aโn = 1/an. It never makes the answer negative: 2โ3 = 1/8.
- A fraction raised to a negative power flips: (2/3)โ2 = (3/2)2 = 9/4.
- A fractional power is a root: a1/2 = โa, a1/3 = โa, and am/n = (โฟโa)m โ do the root first.
- Negative fractional: do all three โ flip, root, power. 8โ2/3 = 1/(โ8)2 = 1/4.
For aโm/n
- Negative sign? Flip the number (write 1 over it).
- Bottom of the fraction? Take that root.
- Top of the fraction? Raise to that power.
Formulas
Worked example
Find the value of (27/8)โ2/3.
- Flip: (8/27)2/3.
- Cube root: โ8 = 2 and โ27 = 3, giving 2/3.
- 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 ร x6 = 27x6 โ the number in front is cubed too.
- 12a5 รท 3aโ2 = 4a5โ(โ2) = 4a7.
- (2x3/5)2 = 4x6/25: square the top and the bottom.
Worked example
Simplify (2x3 y)4 รท 8x5 y2.
- Expand the bracket: 16x12 y4.
- Divide the numbers: 16 รท 8 = 2.
- Subtract powers: x12โ5 = x7, y4โ2 = 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 = 2โ3 and โ2 = 21/2 โ negative and fractional powers often appear here.
- No logarithms are needed (or expected) at this level.
How to do it
- Choose a common base (usually 2, 3 or 5).
- Rewrite each side as basepower, using (am)n = amn.
- Set the powers equal and solve the linear equation.
Worked example
Solve 4x+1 = 8x.
- Base 2: (22)x+1 = (23)x.
- 22x+2 = 23x.
- 2x + 2 = 3x.
x = 2
Exam tip: If one side is a fraction like 1/27, write it as a negative power first: 3โ3.
โCommon mistakes
| Donโt write | Write instead |
|---|---|
| 23 ร 24 = 212 | Add the powers: 27 |
| 23 ร 24 = 47 | The base stays 2: 27 |
| 4โ2 = โ16 | 4โ2 = 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: (โ8)2 = 4 |
โ In the exam
- Non-calculator questions on indices are common โ 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.