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This lesson covers the laws of indices, the meaning of a zero index and of negative indices, square and cube roots, and — for Higher tier [H] — fractional indices. Index notation is a compact way of writing repeated multiplication, and the index laws it obeys run right through the OCR GCSE Mathematics (J560) Number and Algebra strands, including standard form and surds later in this course.
This lesson mainly builds AO1 fluency in applying the index laws, with AO2 reasoning when you explain why a0=1, and AO3 problem-solving in the multi-step evaluations.
| Term | Meaning |
|---|---|
| Base | The number being multiplied repeatedly, the a in an |
| Index (power, exponent) | How many times the base is multiplied by itself |
| Square root | A number that multiplies by itself to give the original; 25=5 |
| Cube root | A number cubed to give the original; 327=3 |
| Reciprocal | a−1=a1 |
| Index law | A rule for combining powers, e.g. am×an=am+n |
Before the rules, be clear on what the notation says. The expression an means "a multiplied by itself n times". So 24 means 2×2×2×2=16, and 53 means 5×5×5=125. The number a is the base and n is the index (also called the power or exponent). Reading an index correctly is half the battle: 32 is "three squared" =9, not "three times two" =6 — a slip that loses easy marks on Paper 1.
Once you see indices as shorthand for repeated multiplication, the index laws stop being something to memorise blindly and become obvious. For example, 23×24 is (2×2×2)×(2×2×2×2), which is 2 multiplied by itself 3+4=7 times, i.e. 27. That is exactly the multiplication law below.
When the base is the same, powers combine by simple rules. Each one comes directly from counting how many times the base is multiplied.
| Law | Rule | Example |
|---|---|---|
| Multiplication | am×an=am+n | 23×24=27 |
| Division | am÷an=am−n | 56÷52=54 |
| Power of a power | (am)n=amn | (32)4=38 |
The division law works the same way in reverse: 5256 has six 5s on top and two on the bottom, so two cancel and four remain, giving 54. The power-of-a-power law comes from "(32) four times", which is 32×32×32×32=32+2+2+2=38 — so you multiply the indices.
OCR Exam Tip: The index laws only apply when the base is the same. You cannot simplify 23×52 with these rules because the bases (2 and 5) differ — you would just work it out as 8×25=200.
Simplify a5×a3÷a2.
Add then subtract the indices: a5+3−2=a6.
Answer: a6.
Simplify 4x36x5×2x4.
Numbers: 46×2=3. Powers of x: x5+4−3=x6.
Answer: 3x6.
Common error: Multiplying the indices in a5×a3 to get a15. You add indices when multiplying powers of the same base.
Following the division law, an÷an=a0; but anything divided by itself is 1, so:
a0=1(for a=0).
A negative index means "reciprocal":
a−n=an1.
Work out (a) 70, (b) 3−2, (c) (52)−1.
(a) 70=1. (b) 3−2=321=91. (c) The reciprocal of 52 is 25.
Answers: (a) 1, (b) 91, (c) 25.
Common error: Writing 3−2=−9 or −6. A negative index does not make the answer negative; it makes a reciprocal.
A square root undoes squaring; a cube root undoes cubing. Knowing your squares and cubes makes these instant on the non-calculator paper.
| n | n2 | n3 |
|---|---|---|
| 2 | 4 | 8 |
| 3 | 9 | 27 |
| 4 | 16 | 64 |
| 5 | 25 | 125 |
Work out (a) 81, (b) 364, (c) 144.
(a) 9×9=81, so 81=9. (b) 43=64, so 364=4. (c) 122=144, so 144=12.
Answers: (a) 9, (b) 4, (c) 12.
Work out 36+327×2.
By BIDMAS, roots act like indices (do them first), then multiply, then add: 36=6, 327=3, so 6+3×2=6+6=12.
Answer: 12.
Write 49 and 3125 as products, then work out 49×3125.
49=7 (since 72=49) and 3125=5 (since 53=125). So 49×3125=7×5=35.
Answer: 35. Knowing the common squares (1,4,9,16,25,36,49,64,81,100) and cubes (1,8,27,64,125) by heart makes root questions instant on the non-calculator paper.
This section is Higher tier only. A fractional index is a root:
an1=na,anm=(na)m=nam.
Take the root first (the denominator n) to keep the numbers small, then raise to the power (the numerator m).
Work out (a) 2521, (b) 2731, (c) 1643.
(a) 2521=25=5. (b) 2731=327=3. (c) 1643=(416)3=23=8.
Answers: (a) 5, (b) 3, (c) 8.
Work out 8−32.
Deal with each part: the negative means reciprocal, the 31 is a cube root, the 2 is a square.
8−32=8321=(38)21=221=41.
Answer: 41.
Common error: Cubing instead of cube-rooting, or forgetting the reciprocal from the minus sign. Handle the sign, the root, and the power as three separate steps.
When the base is negative, the bracket matters. A negative base raised to an even power gives a positive result; raised to an odd power it stays negative, because the minus signs pair up.
Work out (a) (−3)2, (b) (−3)3, (c) −32.
(a) (−3)2=(−3)×(−3)=9 (two negatives multiply to a positive). (b) (−3)3=(−3)×(−3)×(−3)=−27 (three negatives give a negative). (c) −32 has no bracket, so only the 3 is squared and the minus stays outside: −32=−(3×3)=−9.
Answers: (a) 9, (b) −27, (c) −9.
Common error: Reading −32 as (−3)2. Without a bracket the power binds to the 3 alone, so −32=−9, not +9. This single distinction is a classic Paper 1 discriminator.
OCR Exam Tip: In general, a negative base to an even power is positive and to an odd power is negative — count the factors. Always check whether the question has brackets around the negative.
Simplify a4(2a3)2×a−1.
First (2a3)2=4a6. Numerator: 4a6×a−1=4a5. Divide: a44a5=4a1=4a.
Answer: 4a.
Simplify (3x2y)3.
Raise every factor to the power 3: 33×x2×3×y3=27x6y3.
Answer: 27x6y3.
Common error: Forgetting to cube the 3 as well, giving 3x6y3.
Work out (8116)−43.
The negative index flips the fraction: (1681)43. Take the fourth root of top and bottom, then cube: (416)3(481)3=2333=827.
Answer: 827.
Work out the value of n if 2n=32.
Write 32 as a power of 2: 32=25. So n=5.
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