A Zero Power Equals One for a Reason You Can Derive
Use the three laws of indices, work out what zero, negative and fractional powers must mean, solve equations by equating indices, and simplify expressions whose bases look different.
Why does any number to the power zero equal one?
Because it is the only value that keeps the division law working.
That is a much better answer than because the textbook says so, and you can produce it in two lines.
Start with a division you can do without any new rule:
Now do the same division by the subtraction law :
The two routes must agree, so . Nothing about the number mattered, so for every non-zero .
That is the pattern of this whole chapter. Zero, negative and fractional indices are not extra rules bolted on. Each one is forced by insisting that the three original laws keep holding, and each can be derived in a line or two if you forget it.
The one exception worth knowing. is left undefined, because the argument above needs to divide by , and that is illegal when .
This page covers the ICSE Class 9 Mathematics chapter on indices: the three laws, the meaning of zero, negative and fractional powers, solving equations by equating indices, and simplifying compound expressions with different bases.
That is a much better answer than because the textbook says so, and you can produce it in two lines.
Start with a division you can do without any new rule:
Now do the same division by the subtraction law :
The two routes must agree, so . Nothing about the number mattered, so for every non-zero .
That is the pattern of this whole chapter. Zero, negative and fractional indices are not extra rules bolted on. Each one is forced by insisting that the three original laws keep holding, and each can be derived in a line or two if you forget it.
The one exception worth knowing. is left undefined, because the argument above needs to divide by , and that is illegal when .
This page covers the ICSE Class 9 Mathematics chapter on indices: the three laws, the meaning of zero, negative and fractional powers, solving equations by equating indices, and simplifying compound expressions with different bases.
Formula
What are the three laws of indices and when does each one apply?
Multiply the same base and the indices add; divide and they subtract; raise a power to a power and they multiply.
Two more follow from them and are used constantly:
The condition is the part students skip: the first two laws need the same base. does not simplify at all, because there is nothing to count.
Worked example 1. .
Verify it arithmetically: , and , as required. The index just counts how many s are being multiplied, so five of them beside three of them is eight of them.
Worked example 2. , and .
Check the second on numbers: and , while , as required.
Worked example 3 — a compound term. Simplify .
The outside index reaches every factor inside the bracket, the numeral included. Leaving the untouched and writing is the single most common slip here.
Now the confusion this notation invites. and are different objects. , while . The bracket is not decoration — it says which exponent is applied first.
Two more follow from them and are used constantly:
The condition is the part students skip: the first two laws need the same base. does not simplify at all, because there is nothing to count.
Worked example 1. .
Verify it arithmetically: , and , as required. The index just counts how many s are being multiplied, so five of them beside three of them is eight of them.
Worked example 2. , and .
Check the second on numbers: and , while , as required.
Worked example 3 — a compound term. Simplify .
The outside index reaches every factor inside the bracket, the numeral included. Leaving the untouched and writing is the single most common slip here.
Now the confusion this notation invites. and are different objects. , while . The bracket is not decoration — it says which exponent is applied first.
What do negative and fractional indices actually mean?
A negative index means a reciprocal, and a fractional index means a root. Both are forced by the same laws.
Where the negative index comes from. Apply the subtraction law to :
So , and — a negative index on a fraction simply turns the fraction over.
Where the fractional index comes from. Ask what must be. By the power law,
A number whose square is is , so . The same argument with gives , and in general
Worked example 1. .
Taking the root first is almost always easier. The other order gives — the same answer from a bigger number.
Worked example 2. .
Worked example 3. .
Deal with the minus first by inverting, then the :
Worked example 4 — a decimal index. .
The order that saves time: minus sign first (invert), then denominator (root), then numerator (power). Doing the power before the root leaves you extracting a fifth root of instead of one of .
One boundary case. A fractional index needs a non-negative base when the root is even: is not a real number, while is perfectly fine. Odd roots of negatives are allowed; even roots are not.
Where the negative index comes from. Apply the subtraction law to :
So , and — a negative index on a fraction simply turns the fraction over.
Where the fractional index comes from. Ask what must be. By the power law,
A number whose square is is , so . The same argument with gives , and in general
Worked example 1. .
Taking the root first is almost always easier. The other order gives — the same answer from a bigger number.
Worked example 2. .
Worked example 3. .
Deal with the minus first by inverting, then the :
Worked example 4 — a decimal index. .
The order that saves time: minus sign first (invert), then denominator (root), then numerator (power). Doing the power before the root leaves you extracting a fifth root of instead of one of .
One boundary case. A fractional index needs a non-negative base when the root is even: is not a real number, while is perfectly fine. Odd roots of negatives are allowed; even roots are not.
How do you solve an equation when the unknown is in the index?
Write both sides as powers of the same base, then set the indices equal.
If and is positive and not , then . That single statement turns an exponential equation into an ordinary linear one.
Worked example 1. Solve .
, so and .
Worked example 2. Solve .
, so and .
Check: , as required.
Worked example 3 — different-looking bases. Solve .
Neither side is a power of the other, so push both down to the prime base :
Check: , as required. A fractional answer is not a mistake here — it just says the two bases are not whole-number powers of each other.
Worked example 4 — unknown on both sides. Solve .
Both are powers of :
Check: and , as required.
Worked example 5 — collect first. Solve .
Add the indices before comparing: , so and .
Now the condition that makes the whole method legal. Equating indices needs the base to be positive and not equal to . If the base were , then for every , and there is no unique answer to find. Always reduce to a prime base — , or — and the condition takes care of itself.
If and is positive and not , then . That single statement turns an exponential equation into an ordinary linear one.
Worked example 1. Solve .
, so and .
Worked example 2. Solve .
, so and .
Check: , as required.
Worked example 3 — different-looking bases. Solve .
Neither side is a power of the other, so push both down to the prime base :
Check: , as required. A fractional answer is not a mistake here — it just says the two bases are not whole-number powers of each other.
Worked example 4 — unknown on both sides. Solve .
Both are powers of :
Check: and , as required.
Worked example 5 — collect first. Solve .
Add the indices before comparing: , so and .
Now the condition that makes the whole method legal. Equating indices needs the base to be positive and not equal to . If the base were , then for every , and there is no unique answer to find. Always reduce to a prime base — , or — and the condition takes care of itself.
How do you simplify an expression whose bases do not match?
Break every base into primes, take out the common factor, and only then cancel.
These are the questions that look frightening and finish in three lines.
Worked example 1. Simplify .
The temptation is to cancel term by term, which is illegal across a plus sign. Take it out as a factor instead:
**Check with **: , as required. **The answer has no in it at all, which is the point — the question is really asking you to notice that.
Worked example 2.** Simplify .
Write every term as a multiple of :
**Check with **: , as required.
Worked example 3. Simplify .
**Check with **: , as required.
Worked example 4 — a proof. If , show that .
Call the common value . Then , and . Since ,
The move that unlocked it was raising each equation to the power of the reciprocal of its index, which turned a statement about powers into a statement about bases. Once the bases multiplied to give the third base, the addition law finished the proof.
And a link back to expansions. If , then cubing with and gives , so . The identities from the expansions chapter work on indices unchanged — the letters simply stand for surds now.
These are the questions that look frightening and finish in three lines.
Worked example 1. Simplify .
The temptation is to cancel term by term, which is illegal across a plus sign. Take it out as a factor instead:
**Check with **: , as required. **The answer has no in it at all, which is the point — the question is really asking you to notice that.
Worked example 2.** Simplify .
Write every term as a multiple of :
**Check with **: , as required.
Worked example 3. Simplify .
**Check with **: , as required.
Worked example 4 — a proof. If , show that .
Call the common value . Then , and . Since ,
The move that unlocked it was raising each equation to the power of the reciprocal of its index, which turned a statement about powers into a statement about bases. Once the bases multiplied to give the third base, the addition law finished the proof.
And a link back to expansions. If , then cubing with and gives , so . The identities from the expansions chapter work on indices unchanged — the letters simply stand for surds now.
Exam tip
What layout keeps an indices answer from losing marks?
Convert to prime bases on the very first line, and name the law you use on each following line. Indices questions are short, so the marks sit in the reasoning, not the length.
- **Write as , as , as , as , as and as before doing anything else. Nearly every solvable equation in this chapter uses a prime base
- Keep a bracket around a negative index** until you have dealt with it. is , not — the minus applies to the only
- Take out the smallest power as a factor when terms are added or subtracted. Cancelling term by term across a plus sign is the error this section is built to catch
- For fractional indices go in the order: invert, root, power. It keeps the numbers small and the arithmetic mental
- Substitute a small value to check a simplification. If the answer should be independent of , put or and confirm the number. This takes ten seconds and catches a dropped sign
- For a proof, end with the required statement written out in full, not with a line of algebra that happens to be equivalent
The distinction that costs the most marks. but , because in the second the index binds to the before the minus is applied. When a negative number is being raised to a power, the bracket is part of the mathematics.
- **Write as , as , as , as , as and as before doing anything else. Nearly every solvable equation in this chapter uses a prime base
- Keep a bracket around a negative index** until you have dealt with it. is , not — the minus applies to the only
- Take out the smallest power as a factor when terms are added or subtracted. Cancelling term by term across a plus sign is the error this section is built to catch
- For fractional indices go in the order: invert, root, power. It keeps the numbers small and the arithmetic mental
- Substitute a small value to check a simplification. If the answer should be independent of , put or and confirm the number. This takes ten seconds and catches a dropped sign
- For a proof, end with the required statement written out in full, not with a line of algebra that happens to be equivalent
The distinction that costs the most marks. but , because in the second the index binds to the before the minus is applied. When a negative number is being raised to a power, the bracket is part of the mathematics.
Did you know
Why could zero and negative powers not have been defined any other way?
It is natural to read as an arbitrary convention — after all, multiplying a number by itself no times does not obviously mean anything.
But the value is not free. Suppose someone insisted that . Then the addition law gives
which says every power of every number is zero. One wrong definition destroys the entire system. The same test kills every other candidate: only leaves the addition law standing.
The same argument fixes negative and fractional indices. Once you demand that holds for all the numbers you want to allow in the index, the values of and are completely determined. There is no choice left to make.
That is what mathematicians mean by a definition being forced. The notation started as shorthand for repeated multiplication, which only makes sense for counting numbers. Extending it to zero, to negatives and then to fractions was not a matter of taste — each step had exactly one option that preserved the laws.
And the extension does not stop. The next chapter runs the question backwards: instead of asking what is, it asks *what index turns into *. That question is a logarithm, and every law you meet there is one of the three laws above, read from right to left.
But the value is not free. Suppose someone insisted that . Then the addition law gives
which says every power of every number is zero. One wrong definition destroys the entire system. The same test kills every other candidate: only leaves the addition law standing.
The same argument fixes negative and fractional indices. Once you demand that holds for all the numbers you want to allow in the index, the values of and are completely determined. There is no choice left to make.
That is what mathematicians mean by a definition being forced. The notation started as shorthand for repeated multiplication, which only makes sense for counting numbers. Extending it to zero, to negatives and then to fractions was not a matter of taste — each step had exactly one option that preserved the laws.
And the extension does not stop. The next chapter runs the question backwards: instead of asking what is, it asks *what index turns into *. That question is a logarithm, and every law you meet there is one of the three laws above, read from right to left.
Exam relevance
How do indices prepare you for JEE-level algebra and chemistry?
Indices are foundation work, and unusually wide-reaching foundation: three different JEE subjects lean on them without announcing it.
Where it leads in Mathematics. The immediate successor is Logarithms, which is the same content inverted, and the two together support Sequences and Series (geometric progressions are indices in disguise) and the Binomial Theorem in Class 11. JEE Main questions on exponential and logarithmic equations almost always begin by reducing every term to a common prime base — exactly the first line of your layout here.
Where it leads outside Mathematics. Chemistry uses fractional and negative indices constantly in pH and equilibrium work, where a hydrogen-ion concentration is written as a power of ten and manipulated by these laws. Physics uses them for every unit conversion and every order-of-magnitude estimate. In both, the arithmetic slip is nearly always an index sign, not a concept.
Question types to expect. Reducing a compound expression to a single power; solving an exponential equation, sometimes disguised as a quadratic in ; and proof questions of the kind, which show up in JEE Main as short algebra items. Assertion-reason items like to test the conditions — the base must be positive and not one — rather than the manipulation.
The single trap that costs marks. Treating as if a law applied to it. There is no law for adding powers — only for multiplying, dividing and raising them. Every expression with a plus sign has to be factorised first, and candidates who forget that lose the question in its first line.
Board versus competitive emphasis. ICSE rewards the clean derivation and the stated law; a competitive paper only wants the value, so the premium shifts to recognising , , , and as prime powers instantly. Learn those six by sight and most of these questions become one-liners.
Where it leads in Mathematics. The immediate successor is Logarithms, which is the same content inverted, and the two together support Sequences and Series (geometric progressions are indices in disguise) and the Binomial Theorem in Class 11. JEE Main questions on exponential and logarithmic equations almost always begin by reducing every term to a common prime base — exactly the first line of your layout here.
Where it leads outside Mathematics. Chemistry uses fractional and negative indices constantly in pH and equilibrium work, where a hydrogen-ion concentration is written as a power of ten and manipulated by these laws. Physics uses them for every unit conversion and every order-of-magnitude estimate. In both, the arithmetic slip is nearly always an index sign, not a concept.
Question types to expect. Reducing a compound expression to a single power; solving an exponential equation, sometimes disguised as a quadratic in ; and proof questions of the kind, which show up in JEE Main as short algebra items. Assertion-reason items like to test the conditions — the base must be positive and not one — rather than the manipulation.
The single trap that costs marks. Treating as if a law applied to it. There is no law for adding powers — only for multiplying, dividing and raising them. Every expression with a plus sign has to be factorised first, and candidates who forget that lose the question in its first line.
Board versus competitive emphasis. ICSE rewards the clean derivation and the stated law; a competitive paper only wants the value, so the premium shifts to recognising , , , and as prime powers instantly. Learn those six by sight and most of these questions become one-liners.
Key takeaways
What should you be able to do with indices before moving on?
Indices reward understanding over memory, because every rule here can be re-derived from the first three.
- The three laws: , and — the first two need the same base
- **** because must be , and stays undefined
- **, so a negative index on a fraction turns it over
- — take the root first and the numbers stay small; even roots need a non-negative base
- To solve for an index**, write both sides to the same prime base and equate the indices, valid whenever the base is positive and not
- With terms added or subtracted, factor out the smallest power rather than cancelling term by term
- **Check any simplification by substituting or ** and comparing numbers
The test of whether this chapter has landed is whether you can write down and with no working at all. Try those two now, then invent three of your own in the same shape and see if the answers still arrive in one step.
- The three laws: , and — the first two need the same base
- **** because must be , and stays undefined
- **, so a negative index on a fraction turns it over
- — take the root first and the numbers stay small; even roots need a non-negative base
- To solve for an index**, write both sides to the same prime base and equate the indices, valid whenever the base is positive and not
- With terms added or subtracted, factor out the smallest power rather than cancelling term by term
- **Check any simplification by substituting or ** and comparing numbers
The test of whether this chapter has landed is whether you can write down and with no working at all. Try those two now, then invent three of your own in the same shape and see if the answers still arrive in one step.