B1PureFoundation

Index Laws

Q: Why does anything to the power 0 equal 1?

Because the division law forces it: a^na^n = a^n-n = a^0, and any non-zero number over itself is 1. Defining a^0 = 1 is the only choice that keeps the index laws consistent.

Q: Does a negative index make the answer negative?

No. A negative index means take the reciprocal: a^-n = 1a^n. So 2^-3 = 18, which is positive. The sign of the base, not the index, decides whether the value is negative.

Q: For a^m/n, do I take the root or the power first?

Either order gives the same answer, but taking the n-th root first (the denominator) keeps the numbers small and reduces arithmetic slips. For 8^2/3: cube-root to 2, then square to 4.

Q: Can I add the indices of 2^3 5^2?

No — the bases are different. The rule a^m a^n = a^m+n only applies when the base is the same, because it works by pooling identical factors.

An index is just shorthand for repeated multiplication — and every rule of indices, even the odd-looking ones like $a^{0} = 1$ and $a^{- 1} = \frac{1}{a}$ , comes straight from that one idea. Understand where the laws come from and you never have to memorise them.

25 min Video by TLMaths Algebra and Functions

The big picture

Indices are the language of growth, decay, area, volume, compound interest and every exponential model you will meet later. At A-Level they stop being a topic and become a tool you use everywhere — differentiating, integrating, solving equations, working with logarithms. The students who struggle later are almost always the ones who learned the index laws as disconnected rules to memorise. The students who fly learned them as consequences of a single idea: a power counts how many times you multiply. This lesson makes sure you are the second kind.

A-Level Maths B1-01: The Laws of IndicesWatch the walkthrough, then work through the notes below.

Open on YouTube

What you'll be able to do

Use the three core laws for multiplying, dividing and raising powers
Explain why $a^{0} = 1$ and why a negative index means a reciprocal
Interpret a fractional index as a root, and evaluate expressions like $8^{2/3}$
Simplify expressions with several index laws combined

What a power actually counts

The expression $a^{n}$ means “multiply $a$ by itself $n$ times”. The $a$ is the $base$ and the $n$ is the $index$ (or power, or exponent). So $2^{5} = 2 \times 2 \times 2 \times 2 \times 2 = 32$ .

That single definition is the seed for everything. Hold onto the picture of “counting factors” — the moment a rule looks mysterious, count the factors and it becomes obvious.

The three laws that follow immediately

Because a power is just a count of factors, multiplying two powers of the $same base$ pools their factors, so you add the indices. Dividing cancels factors, so you subtract. And a power of a power repeats the multiplication, so you multiply the indices.

a^{m} \times a^{n} = a^{m + n}

Same base, multiply → add the indices.

\frac{a ^{m}}{a ^{n}} = a^{m - n}

Same base, divide → subtract the indices.

(a^{m})^{n} = a^{mn}

A power of a power → multiply the indices.

1Multiply:

x^{7} \times x^{3} = x^{7 + 3} = x^{10}

2Divide:

x^{10} \div x^{4} = x^{10 - 4} = x^{6}

Answer

x^{6}

Why this works

Why must the bases match to add indices? Because $2^{3} \times 5^{2}$ pools threes with fives — there is no single base whose factor-count you can add. The rule $a^{m} \times a^{n} = a^{m + n}$ only works when the thing being multiplied is the same each time.

Tip — The laws add/subtract indices for $\times$ and $\div$ — never for $+$ or $-$ . $a^{3} + a^{2}$ does not simplify to $a^{5}$ ; there is nothing to combine.

The rules that are forced: zero and negative indices

What could $a^{0}$ possibly mean — multiplying $a$ by itself zero times? Rather than guess, we demand that the laws keep working. By the division law, $\frac{a ^{n}}{a ^{n}} = a^{n - n} = a^{0}$ . But any non-zero number divided by itself is $1$ . So $a^{0}$ is forced to equal $1$ .

The same logic pins down negative indices. Since $\frac{a ^{3}}{a ^{5}} = a^{3 - 5} = a^{- 2}$ , and cancelling factors leaves $\frac{1}{a ^{2}}$ , a negative index must mean “one over”. A minus sign in the index is an instruction to take the reciprocal — it never makes the number negative.

a^{0} = 1 (a \neq = 0)

Forced by the division law.

a^{- n} = \frac{1}{a ^{n}}

A negative index means reciprocal — not a negative answer.

5^{0} = 1

by definition.

2^{- 3} = \frac{1}{2 ^{3}} = \frac{1}{8}

Answer

5^{0} = 1

and

2^{- 3} = \frac{1}{8}

Why this works

Notice the strategy: we did not invent meanings for $a^{0}$ and $a^{- n}$ arbitrarily. We chose the only values that keep the three laws consistent. Good mathematics extends rules by preserving patterns, not by decree.

Fractional indices are roots

A fractional index continues the same story. What is $a^{1/2}$ ? Whatever it is, the power law says $(a^{1/2})^{2} = a^{1} = a$ . So $a^{1/2}$ is the number that squares to give $a$ — the square root. In general $a^{1/ n}$ is the $n$ -th root.

A fraction $\frac{m}{n}$ in the index does two jobs at once: the bottom takes a root, the top raises to a power. It is usually easier to take the root first (the numbers stay small).

a^{\frac{1}{n}} = n a

The denominator is the root.

a^{\frac{m}{n}} = (n a)^{m}

Root first (bottom), then power (top).

1Root first:

38 = 2

2Then power:

2^{2} = 4

Answer

8^{\frac{2}{3}} = 4

Tip — Turning roots and reciprocals into index form (like $x^{- 1/2}$ ) is essential before you differentiate or integrate — it is one of the most useful skills in the whole of Pure.

Think like an examiner

“Simplify” or “write in the form $a x^{n}$ ” means every root and fraction must become a single index — leave no or fraction bars in the final power.
A negative index is a favourite trap: $4^{- 1/2} = \frac{1}{2}$ , not $- 2$ . The sign controls reciprocal, the fraction controls the root.
When a bracket is raised to a power, apply it to the coefficient too: $(2 x)^{3} = 8 x^{3}$ , a mark routinely lost by writing $2 x^{3}$ .
Show the split of a fractional power (root, then power) — method marks are available even if the arithmetic slips.

Common misconceptions

Writing

a^{3} + a^{4} = a^{7}

Indices only add when powers are multiplied. Addition of powers does not simplify.

Thinking

2^{- 3} = - 8

2^{- 3} = \frac{1}{2 ^{3}} = \frac{1}{8}

. A negative index means reciprocal, not a negative value.

(2 x)^{3} = 2 x^{3}

The power hits the whole bracket:

(2 x)^{3} = 2^{3} x^{3} = 8 x^{3}

Evaluating

8^{2/3}

8^{2} \div 3

or similar.

8^{2/3} = (38)^{2} = 2^{2} = 4

: cube root then square.

The index laws

a^{m} \times a^{n} = a^{m + n}

Multiply → add.

a^{m} \div a^{n} = a^{m - n}

Divide → subtract.

(a^{m})^{n} = a^{mn}

Power of a power → multiply.

a^{0} = 1

Zero index.

a^{- n} = \frac{1}{a ^{n}}

Negative → reciprocal.

a^{\frac{m}{n}} = (n a)^{m}

Fractional → root then power.

Stretch yourself

Solve for $x$ : $2^{x} \times 4^{x + 1} = 8^{2}$ . (Hint: write every term as a power of 2.)

Hint — Use $4 = 2^{2}$ and $8 = 2^{3}$ , then apply the laws so both sides are a single power of 2 and equate the indices.

Questions students ask

Why does anything to the power 0 equal 1?

Because the division law forces it:

\frac{a ^{n}}{a ^{n}} = a^{n - n} = a^{0}

, and any non-zero number over itself is 1. Defining

a^{0} = 1

is the only choice that keeps the index laws consistent.

Does a negative index make the answer negative?

No. A negative index means take the reciprocal:

a^{- n} = \frac{1}{a ^{n}}

. So

2^{- 3} = \frac{1}{8}

, which is positive. The sign of the base, not the index, decides whether the value is negative.

For

a^{m / n}

, do I take the root or the power first?

Either order gives the same answer, but taking the

n

-th root first (the denominator) keeps the numbers small and reduces arithmetic slips. For

8^{2/3}

: cube-root to 2, then square to 4.

Can I add the indices of

2^{3} \times 5^{2}

No — the bases are different. The rule

a^{m} \times a^{n} = a^{m + n}

only applies when the base is the same, because it works by pooling identical factors.

Key takeaways

A power counts repeated multiplication — every law follows from that.
Multiply → add indices; divide → subtract; power of a power → multiply.
$a^{0} = 1$ and $a^{- n} = \frac{1}{a ^{n}}$ are forced by keeping the laws consistent.
A fractional index is a root: $a^{m / n} = (n a)^{m}$ — root first, then power.
Rewriting roots/reciprocals as indices is the key setup step for calculus.

How this fits the course

Build on

Indices and roots (GCSE)soon

Surds and Rationalising

Leads to

Surds and Rationalising
Differentiating xⁿ (later)soon

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