Indices, Radicals and Rational Expressions
The Index Laws: a Complete Treatment
Why powers add on multiplication and multiply on nesting, how those two facts force the values of x0 and x-m, and why 00 is left undefined.
What this page covers
- State and apply the three index laws for multiplication, division and nesting
- Derive the value of a0 rather than memorising it
- Explain what a negative exponent means and where it comes from
- Say why 00 has no consistent value
What a power abbreviates
A positive whole-number exponent is shorthand for repeated multiplication, and nothing more. The source notes begin exactly there, because every law that follows can be read off the shorthand.
- Base
- The quantity being multiplied — in 23 it is 2
- Exponent (index, power)
- How many times — in 23 it is 3
- a1
- Equals a. One factor is still a product
Everything on this page for whole-number exponents follows by counting factors. The value of the three laws is that they continue to hold once the exponent is zero, negative or fractional, where counting factors no longer means anything — and that continuation is what defines those cases.
Law 1: multiplying powers of the same base
The source derives it by counting. 23 × 24 is three 2s multiplied by four 2s, which is seven 2s in all.
The bases must match. 23 × 34 does not simplify by this law; there is no way to combine different bases into a single power.
Law 2: dividing powers of the same base
Counting again: 2522 has five 2s on top and two below, and two of them cancel, leaving three.
The source deliberately writes division as multiplication by a negative power. That is the right instinct: it means there is really only one law here, not two, and the same arithmetic covers both.
Law 3: raising a power to a power
Counting once more: (22)3 means three copies of 22 multiplied together, which is 22 × 22 × 22 = 26.
(am)n and amn are different. The first multiplies the exponents and equals amn; the second raises a to the power mn. Compare (22)3 = 26 = 64 against 223 = 28 = 256. Brackets are not decoration here.
Two further laws, for products and quotients
(a + b)n is not an + bn. This is the single most frequently repeated error in elementary algebra. At a = b = 1 and n = 2 the true value is 4 and the mistaken one is 2. The correct expansion is in Multiplying Polynomials and the Special Products.
The zero exponent, derived
Counting factors gives no meaning to a0 — zero copies of anything is not a product. The value is instead forced by insisting that Law 1 continue to hold.
Deriving a0 = 1
So 20 is a number that multiplies 23 and leaves it unchanged. The only such number is 1.
Nothing about the base 2 was used, so the same argument gives a0 = 1 for every non-zero a. The source notes record (-4)0 = 1 to make the point that negative bases are included.
Why 00 is left undefined
The derivation above divides by a3 in effect, and that is illegitimate when a = 0. Two competing patterns then give different answers.
| Pattern | Suggests | Evidence |
|---|---|---|
| a0 = 1 for every non-zero a | 00 = 1 | 0.10 = 1, 0.010 = 1, and so on |
| 0n = 0 for every positive n | 00 = 0 | 00.1 = 0, 00.01 = 0, and so on |
Neither pattern has a claim on priority, and approaching the point along different paths genuinely produces different limits. The source notes therefore record 00 as not defined, which is the standard position in analysis.
Negative exponents, derived
The same technique fixes the meaning of a negative exponent. The source works it concretely first.
Deriving 2-3 = 18
Equivalently, a-m is the multiplicative inverse of am, which is exactly what Law 1 requires: a3 × a-3 = a0 = 1. The source checks this directly, writing a3 × 1a3 = 1.
| Expression | Equals | Note |
|---|---|---|
| 4y-2 | 4y2 | Source example. Only y carries the negative power; the 4 stays put |
| 1x-3 | x3 | A negative power in the denominator moves up |
| 2x-13y-2 | 2y23x | Each negative-power factor crosses the bar |
| (3x)-2 | 19x2 | The exponent applies to the whole bracket, including the 3 |
4y-2 is 4y2, not 14y2 and not 1(4y)2. A negative exponent attaches to the factor it is written on and to nothing else. When in doubt, insert brackets to make the scope explicit.
The laws in one table
| Law | Statement | Condition |
|---|---|---|
| Multiplication | aman = am+n | Same base |
| Division | aman = am-n | Same base, a ≠ 0 |
| Nesting | (am)n = amn | — |
| Product | (ab)n = anbn | — |
| Quotient | (ab)n = anbn | b ≠ 0 |
| Zero | a0 = 1 | a ≠ 0 |
| Negative | a-m = 1am | a ≠ 0 |
Every one of these was derived, not asserted. That matters: when a case arises that is not on the list, the way to resolve it is to ask what value keeps the laws consistent, which is precisely how a0 and a-m were settled.
Worked practice
Simplify (2x3y-1)24x-1y3
- Deal with the bracket first. The exponent 2 distributes over every factor: (2x3y-1)2 = 4x6y-2.
- Write the quotient. 4x6y-24x-1y3.
- Subtract exponents, base by base. The 4s cancel; x6-(-1) = x7; y-2-3 = y-5.
- Clear the negative power. x7y-5 = x7y5.
At x = y = 2: the original is (2 · 8 · 12)24 · 12 · 8 = 6416 = 4, and 2725 = 4. Agreement.
Two conventions are worth adopting. Clear all negative exponents from a final answer unless the question asks otherwise, and handle brackets before quotients — distributing an exponent over a bracket first prevents the most common scope error.
Common mistakes
| Mistake | Correct | Counterexample |
|---|---|---|
| am · an = amn | am+n | 22 · 23 = 32, not 26 = 64 |
| (am)n = am+n | amn | (22)3 = 64, not 25 = 32 |
| (a + b)2 = a2 + b2 | a2 + 2ab + b2 | a = b = 1: 4, not 2 |
| a0 = 0 | a0 = 1 for a ≠ 0 | Otherwise 23 · 20 would be 0 |
| a-m = -am | a-m = 1am | 2-3 = 18, not -8 |
| 4y-2 = 14y2 | 4y2 | The exponent attaches to y only |
| 23 · 34 = 67 | No simplification | 8 · 81 = 648, but 67 is 279936 |
Frequently asked questions
Why is a0 = 1 rather than 0?
Because it is forced. 23 · 20 = 23 + 0 = 23, so 20 must be whatever multiplies 23 and leaves it unchanged — that is 1. Defining it as 0 would break the multiplication law for every base.
Is (-4)0 also 1?
Yes. The same argument works for any non-zero base, positive or negative. The source notes state (-4)0 = 1 explicitly.
Why is 00 undefined?
Two rules disagree. 'Anything to the power 0 is 1' gives 1; 'zero to any power is 0' gives 0. Neither has priority, and in limits both values genuinely arise depending on how the base and exponent approach zero. Leaving it undefined is the only consistent choice.
Does (a + b)2 equal a2 + b2?
No. The index laws govern products and quotients, never sums. (ab)2 = a2b2 is correct; (a + b)2 = a2 + 2ab + b2, with a middle term that must not be dropped.
Source. Handwritten teaching notes, Week 2, pages 1-4.
This page is an original exposition prepared for the KEVOS® knowledge library. It restates, reorganises and verifies the mathematics of the supplied teaching notes; it is not a reproduction of them. Numerical values taken from the notes are identified as source examples and are not presented as engineering standards.