Why is "Multiplying the indices when multiplying powers, e.g. $2^3 \times 2^4 = 2^{12}$." a common mistake in Power and Roots?

Why it happens: Multiplication of the powers gets mixed up with multiplication of the indices. How to avoid it: Multiplying powers adds the indices: 2^3 × 2^4 = 2^3+4 = 2^7.

Why is "Reading a negative index as a negative answer, e.g. $2^{-3} = -8$." a common mistake in Power and Roots?

Why it happens: The minus sign in the index is assumed to make the whole value negative. How to avoid it: A negative index means a reciprocal: 2^-3 = 1/2^3 = 1/8 — a positive fraction.

Why is "Thinking $a^0 = 0$ instead of $a^0 = 1$." a common mistake in Power and Roots?

Why it happens: An index of zero looks like it should produce an answer of zero. How to avoid it: The dividing law gives a^n ÷ a^n = a^0, and a number over itself is 1, so a^0 = 1.

Why is "Using an index law on different bases, e.g. simplifying $2^3 \times 5^2$." a common mistake in Power and Roots?

Why it happens: The laws are remembered as 'add the indices' without the same-base condition. How to avoid it: The index laws only work when the base is the same. 2^3 × 5^2 cannot be combined.

Why is "Treating $3^4$ as $3 \times 4 = 12$." a common mistake in Power and Roots?

Why it happens: The index is read as a multiplier rather than a count of factors. How to avoid it: The index counts factors, so 3^4 = 3 × 3 × 3 × 3 = 81.

The three laws of indices at a glance

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Powers and Roots — Cambridge Lower Secondary Maths, Grade 8

You already know the three index laws. Stage 9 stretches them — a power of zero, negative indices that mean fractions, and using every law together inside longer simplifications without losing track of a single index.

At a glance

Multiplying powers of the same base adds the indices: $a^m \times a^n = a^{m+n}$ .
Dividing powers of the same base subtracts the indices: $a^m \div a^n = a^{m-n}$ .
Raising a power to a power multiplies the indices: $(a^m)^n = a^{mn}$ .
Any non-zero number to the power 0 equals 1, so $a^0 = 1$ .
A negative index means a reciprocal: $a^{-n} = \frac{1}{a^n}$ .
The index laws only ever work when the base is the same.
A square root undoes squaring; a cube root undoes cubing.
Estimate a root by trapping it between two perfect squares or cubes.

What you’ll learn

Mapped to the Cambridge Lower Secondary Mathematics curriculum framework.

Stage 9 — Apply all three laws of indices fluently within longer simplifications.
Stage 9 — Use the zero index, explaining why any non-zero number to the power 0 is 1.
Stage 9 — Use negative indices and convert between $a^{-n}$ and $\frac{1}{a^n}$ .

The index laws, used fluently

All three laws should now run together inside a single longer simplification.

By Stage 9 the three laws of indices should be second nature. The new skill is using them together — a single expression often needs more than one law before it is fully simplified.

The three laws, all needing the same base:

Multiplying: $a^m \times a^n = a^{m+n}$ — add the indices.

The zero index

Any non-zero number raised to the power 0 equals 1 — and the dividing law shows why.

What does a power of zero mean? You cannot multiply a number "zero times", so the meaning comes from the dividing law instead.

Think about $a^4 \div a^4$ . A number divided by itself is $1$ . But the dividing law says $a^{4} \div$ . Both lines describe the same calculation, so must equal .

Negative indices

A negative index means a reciprocal — flip the base under 1.

Carry the halving pattern past zero. After $2^1 = 2$ and $2^0 = 1$ , dividing by 2 again gives , then . A means a .

Roots and estimating them

Roots undo powers — trap an awkward root between two perfect values.

A root undoes a power. The square root asks "what was squared?" — since $9^2 = 81$ , $\sqrt{81} = 9$ . The asks "what was cubed?" — since , .

Combining the laws in one go

Harder questions chain several laws together — work one step at a time.

The toughest Stage 9 index questions ask you to simplify an expression that needs several laws in turn. The cure is the same as always: one step per line.

Try $\frac{(2^3)^2 \times 2^4}{2^{12}}$ .

Where you'll use this next

Index fluency, especially negative indices, underpins standard form and algebra.

Confident work with powers and roots opens up a great deal of later maths:

Standard form depends entirely on powers of 10, and small numbers use the negative indices you have just met.
Algebra uses the index laws constantly — simplifying $x^5 \div x^2$ to $x^3$ is the very same rule you used with numbers.

Quick recap

Multiplying powers of the same base adds the indices.
Dividing powers of the same base subtracts the indices.
Raising a power to a power multiplies the indices.
Any non-zero number to the power 0 equals 1.
A negative index means a reciprocal: $a^{-n} = \frac{1}{a^n}$ .

Sources: Cambridge Lower Secondary Mathematics curriculum framework (Stage 9). Last reviewed 2026-05-19.

a^{4}

=

a^{4 - 4}

=

a^{0}

a^4 \div a^4 = a^{4-4} = a^0

2^{- 1}

=

\frac{1}{2}

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

7^{2} = 49

7^2 = 49

Power and Roots — Cambridge Lower Secondary Mathematics — Stage 9 Revision Notes & Practice | Tutopiya

Power and Roots

Detailed Study Notes

Take this whole topic with you

Powers and Roots — Cambridge Lower Secondary Maths, Grade 8

At a glance

What you’ll learn

The index laws, used fluently

The zero index

Negative indices

Roots and estimating them

Combining the laws in one go

Where you'll use this next

Quick recap

Step-by-step worked examples — Power and Roots

1Multiplying and dividing powers

2Using the zero index

3Turning a negative index into a fraction

4Estimating a square root

5Combining the index laws

Key Definitions and Keywords — Power and Roots

Power

Base

Index (exponent)

Laws of indices

Zero index

Negative index

Reciprocal

Square root

Cube root

Perfect square

Estimate

Power of 10

Common Mistakes and Misconceptions — Power and Roots

✕Multiplying the indices when multiplying powers, e.g. $2^3 \times 2^4 = 2^{12}$ .

✕Reading a negative index as a negative answer, e.g. $2^{-3} = -8$ .

✕Thinking $a^0 = 0$ instead of $a^0 = 1$ .

✕Using an index law on different bases, e.g. simplifying $2^3 \times 5^2$ .

✕Treating $3^4$ as $3 \times 4 = 12$ .

Power and Roots

Detailed Study Notes

Take this whole topic with you

At a glance

What you’ll learn

Quick recap

1Multiplying and dividing powers

2Using the zero index

3Turning a negative index into a fraction

4Estimating a square root

5Combining the index laws

Power

Base

Index (exponent)

Laws of indices

Zero index

Negative index

Reciprocal

Square root

Cube root

Perfect square

Estimate

Power of 10

✕Multiplying the indices when multiplying powers, e.g. 23×24=2122^3 \times 2^4 = 2^{12}23×24=212.

✕Reading a negative index as a negative answer, e.g. 2−3=−82^{-3} = -82−3=−8.

✕Thinking a0=0a^0 = 0a0=0 instead of a0=1a^0 = 1a0=1.

✕Using an index law on different bases, e.g. simplifying 23×522^3 \times 5^223×52.

✕Treating 343^434 as 3×4=123 \times 4 = 123×4=12.

✕Multiplying the indices when multiplying powers, e.g. $2^3 \times 2^4 = 2^{12}$ .

✕Reading a negative index as a negative answer, e.g. $2^{-3} = -8$ .

✕Thinking $a^0 = 0$ instead of $a^0 = 1$ .

✕Using an index law on different bases, e.g. simplifying $2^3 \times 5^2$ .

✕Treating $3^4$ as $3 \times 4 = 12$ .