Why is "Leaving the number part outside the 1-to-10 range, e.g. $25 \times 10^6$." a common mistake in Standard Form?

Why it happens: The calculation is finished without the final check on a. How to avoid it: Make sure a is at least 1 and less than 10: 25 × 10^6 = 2.5 × 10^7.

Why is "Thinking a negative power makes the number negative, e.g. $3 \times 10^{-2} = -300$." a common mistake in Standard Form?

Why it happens: The minus sign in the power is read as a minus on the whole value. How to avoid it: A negative power means a small positive number: 3 × 10^-2 = 0.03.

Why is "Multiplying the powers of 10 when multiplying standard-form numbers." a common mistake in Standard Form?

Why it happens: The two operations — multiplying numbers and combining powers — get mixed up. How to avoid it: Multiplying powers of 10 adds the indices: 10^4 × 10^5 = 10^9.

Why is "Counting the wrong number of places when moving the decimal point." a common mistake in Standard Form?

Why it happens: Zeros are skipped or double-counted while moving the point. How to avoid it: Move the point one digit at a time and tally each move; check against the digit count.

Why is "Ordering standard-form numbers by their number part instead of the power." a common mistake in Standard Form?

Why it happens: The number part is read first, so it feels like the deciding value. How to avoid it: Compare the powers of 10 first; only use the number part when the powers are equal.

Converting a large number to standard form

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Standard Form — Cambridge Lower Secondary Maths, Grade 8

Some numbers are far too big or too small to write out comfortably. Standard form is a neat shorthand built on powers of 10 — this guide shows you how to write numbers in it and calculate with them confidently.

At a glance

Standard form writes a number as $a \times 10^n$ .
The number $a$ must be at least 1 and less than 10.
A large number has a positive power of 10.
A small number, less than 1, has a negative power of 10.
The power $n$ counts how many places the decimal point moves.
To multiply standard-form numbers, multiply the $a$ values and add the powers.
To divide, divide the $a$ values and subtract the powers.
Always tidy the answer so $a$ is between 1 and 10.

What you’ll learn

Mapped to the Cambridge Lower Secondary Mathematics curriculum framework.

Stage 9 — Write large and small numbers in standard form $a \times 10^n$ .
Stage 9 — Convert numbers between standard form and ordinary form.
Stage 9 — Order and compare numbers written in standard form.
Stage 9 — Multiply and divide numbers written in standard form.

What standard form is

Standard form writes any number as a number between 1 and 10 times a power of 10.

Some numbers are awkward to write out in full. The distance to the Sun is about $150\,000\,000$ km; the width of a human hair is about $0.000\,07$ m. Counting all those zeros is slow and easy to get wrong.

Standard form is a tidy shorthand. Every number is written as $a \times 10^n,$ where is , and is an integer.

Large numbers in standard form

A big number has a positive power of 10 — count the places the point moves.

For a large number — one bigger than 10 — the power of 10 is positive.

The method: place a decimal point so that exactly one non-zero digit sits in front of it, then count how many places the point has moved. That count is the power.

The point slides left; the number of places is the power.

For $4\,800\,000$ : place the point after the 4 to get $4.8$ . The point has moved 6 places, so the answer is .

Small numbers in standard form

A number less than 1 has a negative power of 10.

For a small number — one between 0 and 1 — the power of 10 is negative. This is where the negative indices you have met come into their own.

The method is the mirror image. Move the decimal point right until one non-zero digit sits in front of it, count the places, and make the power negative.

The point slides right; the power of 10 is negative.

For $0.000\,36$ : move the point right until it sits after the 3, giving $3.6$ . The point moved 4 places, so the answer is .

Comparing and ordering

Compare the powers of 10 first; only use the number part to break a tie.

Standard form makes ordering numbers wonderfully quick — far quicker than counting zeros.

The method has two steps. First, compare the powers of 10 — the bigger power means the bigger number. Only if two numbers share the same power do you look at the number part $a$ to break the tie.

Compare powers first; the number part only breaks a tie.

So $8 \times 10^3$ , and in order are: first (smallest power), then and — same power, so 2 comes before 9.

Multiplying and dividing in standard form

Handle the number parts and the powers of 10 separately, then tidy.

One of the best things about standard form is how neatly it multiplies and divides — you deal with the number parts and the powers separately.

To multiply: multiply the $a$ values, and add the powers (the multiplying index law).
To divide: divide the $a$ values, and subtract the powers.

Number parts multiply; powers of 10 add.

So $($ .

Where you'll use this next

Standard form is the everyday language of science and large data.

Standard form is used far beyond the maths classroom:

Science writes almost every measurement in standard form — the mass of an atom, the speed of light, the distance to a star.
Calculators display very large or very small results in standard form, so reading it fluently is a real-world skill.
Algebra in later stages uses the same index laws on powers of 10 that you have practised here.
Computing and data describe storage and processing speeds with the same shorthand, such as $5 \times 10^9$ operations.

If standard form feels shaky later, it is usually the rule on $a$ , or an index law for the powers, that needs a refresh. The "tidy into range" habit is the one to lock in.

Quick recap

Standard form is $a \times 10^n$ , with $a$ at least 1 and less than 10.
A large number has a positive power of 10.
A small number, less than 1, has a negative power of 10.
The power counts how many places the decimal point moves.
Compare powers of 10 first when ordering; the number part breaks ties.
Multiply: multiply the $a$ values and add the powers.
Divide: divide the values and subtract the powers, then tidy.

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

3.6 \times 10^{-4}

3

\times

10^{4}

)

\times

(

2

\times

10^{5}

)

=

(

3

\times

2

)

\times

10^{4 + 5}

=

6

\times

10^{9}

(3 \times 10^4) \times (2 \times 10^5) = (3 \times 2) \times 10^{4+5} = 6 \times 10^9

a

Standard Form — Cambridge Lower Secondary Mathematics — Stage 9 Revision Notes & Practice | Tutopiya

Standard Form

Detailed Study Notes

Take this whole topic with you

Standard Form — Cambridge Lower Secondary Maths, Grade 8

At a glance

What you’ll learn

What standard form is

Large numbers in standard form

Small numbers in standard form

Comparing and ordering

Multiplying and dividing in standard form

Where you'll use this next

Quick recap

Step-by-step worked examples — Standard Form

1Writing a large number in standard form

2Converting back to an ordinary number

3Writing a small number in standard form

4Ordering numbers in standard form

5Multiplying and tidying the answer

Key Definitions and Keywords — Standard Form

Standard form

Number part

Power of 10

Index (power)

Positive power

Negative power

Ordinary form

Scientific notation

Decimal point

Place value

Integer

Tidy up (standard form)

Common Mistakes and Misconceptions — Standard Form

✕Leaving the number part outside the 1-to-10 range, e.g. $25 \times 10^6$ .

✕Thinking a negative power makes the number negative, e.g. $3 \times 10^{-2} = -300$ .

✕Multiplying the powers of 10 when multiplying standard-form numbers.

✕Counting the wrong number of places when moving the decimal point.

✕Ordering standard-form numbers by their number part instead of the power.

Standard Form

Detailed Study Notes

Take this whole topic with you

At a glance

What you’ll learn

Quick recap

1Writing a large number in standard form

2Converting back to an ordinary number

3Writing a small number in standard form

4Ordering numbers in standard form

5Multiplying and tidying the answer

Standard form

Number part

Power of 10

Index (power)

Positive power

Negative power

Ordinary form

Scientific notation

Decimal point

Place value

Integer

Tidy up (standard form)

✕Leaving the number part outside the 1-to-10 range, e.g. 25×10625 \times 10^625×106.

✕Thinking a negative power makes the number negative, e.g. 3×10−2=−3003 \times 10^{-2} = -3003×10−2=−300.

✕Multiplying the powers of 10 when multiplying standard-form numbers.

✕Counting the wrong number of places when moving the decimal point.

✕Ordering standard-form numbers by their number part instead of the power.

✕Leaving the number part outside the 1-to-10 range, e.g. $25 \times 10^6$ .

✕Thinking a negative power makes the number negative, e.g. $3 \times 10^{-2} = -300$ .