GCSE · Maths · AQA · Spec 8300

Standard form

Try reading this out loud: 3 400 000 000 000. Counting zeros in threes? Everyone does. Standard form fixes that — and makes huge and tiny numbers easy to compare.

Where does it really sit?

Which is bigger: 9 × 10³ or 1.1 × 10⁴?

Each number is written as A × 10ⁿ: a number that is at least 1 but less than 10, times a power of 10. Remember 10³ = 1000 and 10⁴ = 10 000. Drag each slider to where you think the number sits, then check.

P — 9 × 10³Q — 1.1 × 10⁴R — 4 × 10³S — 2.5 × 10⁴T — 3.6 × 10⁴

Is it really standard form?

Sort each number, then fix the ones that break the rule

Pick a number, then pick the box it belongs in. Read the reason each time — that's where the fix is.

Still to sort

Already in standard form (0)

A is at least 1 and less than 10, and the index is an integer.

Where the line is: 1 is allowed as A; 10 is not. So 1 × 10⁸ is fine, but 10 × 10⁻³ is not.

A is 10 or more (0)

Make A smaller and the index bigger by the same number of tens.

Where the line is: Exactly 10 counts as too big: A must be LESS than 10.

A is less than 1 (0)

Make A bigger and the index smaller by the same number of tens.

Where the line is: Anything from 1 upwards is fine — it is only below 1 that A needs fixing.

Index is not an integer (0)

The index has to be a whole number: positive, negative or zero.

8 of 8 still to sort.

The rule: A × 10ⁿ with 1 ≤ A < 10 and n an integer.

Multiplying and dividing

Problem

Work out, giving each answer in standard form: (a) (6 × 10⁴) × (5 × 10³) (b) (2.4 × 10⁶) ÷ (8 × 10²)

Now for adding

You just added indices to multiply. Does that work for adding too?

Work out (3 × 10⁴) + (5 × 10³).

Which is closest to what you would do?

Add the A numbers and add the indices: 3 + 5 = 8 and 4 + 3 = 7, so 8 × 10⁷.Add the A numbers and keep the bigger index: 3 + 5 = 8, so 8 × 10⁴.Line them up by place value: 30 000 + 5000 = 35 000, which is 3.5 × 10⁴.

How sure are you?

I'm sureI think soI'm guessing

Add the A numbers and add the indices: 3 + 5 = 8 and 4 + 3 = 7, so 8 × 10⁷.

This is a common misconception

That's the multiplication rule sneaking in. Adding indices is what happens when you MULTIPLY powers of 10 — this is an addition. Write the numbers out and line them up in place-value columns: 3 × 10⁴ is 30 000, with its 3 in the ten-thousands column; 5 × 10³ is 5000, with its 5 in the thousands column. Adding just drops each digit into its own column: 35 000. Nothing got anywhere near 10⁷ — 8 × 10⁷ is 80 000 000.

Add the A numbers and keep the bigger index: 3 + 5 = 8, so 8 × 10⁴.

This is a common misconception

You're right that the powers shouldn't be added — but the 5 isn't 5 ten-thousands. In place-value columns, the 5 from 5 × 10³ sits in the thousands column, one column to the right of the 3. You can only add A numbers directly when their powers of 10 match. Rewrite 5 × 10³ as 0.5 × 10⁴; then 3 + 0.5 = 3.5, giving 3.5 × 10⁴. Your 8 × 10⁴ is 80 000, which is far too big.

Line them up by place value: 30 000 + 5000 = 35 000, which is 3.5 × 10⁴.

This matches the evidence

Exactly. In the place-value columns, the 3 stays in the ten-thousands column and the 5 stays in the thousands column — adding doesn't move either of them. Then rewrite 35 000 in standard form: 3.5 × 10⁴. You can also do it without writing out the zeros: make the powers match first (5 × 10³ = 0.5 × 10⁴) and add the A numbers.

Multiplying and dividing combine the powers of 10. Adding and subtracting never do — every digit keeps its place value. So before you add or subtract, get both numbers to the same power of 10 (or write them out in full).

Your turn to fill the gaps

Adding and subtracting with different powers

Work out, giving each answer in standard form: (a) (9.5 × 10⁵) + (7 × 10⁴) (b) (1.5 × 10⁻³) − (8 × 10⁻⁴)

(a) 10⁵ and 10⁴ don't match yet. Rewrite the smaller number with 10⁵.
missing step

Which line is step 2?

7 × 10⁴ = 0.7 × 10⁵7 × 10⁴ = 7 × 10⁵7 × 10⁴ = 70 × 10⁵

Put it all together

One without a calculator, one with

(a) Without a calculator, work out (5 × 10⁻³) × (4 × 10⁸). Give your answer in standard form. (b) Mia uses her calculator to work out 0.0094 ÷ 200. Her calculator display shows: 4.7 × 10⁻⁰⁵ (i) Write Mia's answer in standard form. (ii) Write Mia's answer as an ordinary number. [4 marks]

Your answer

0 words · your answer stays on this page and is not sent anywhere.

The mark scheme

(a) Multiplies the A numbers and adds the indices: 5 × 4 = 20 and 10⁻³ × 10⁸ = 10⁵ (for example 20 × 10⁵ seen) (1 mark)
(a) 2 × 10⁶ (1 mark)
(b)(i) 4.7 × 10⁻⁵ (1 mark)
(b)(ii) 0.000047 (1 mark)

Model answer

(a) (5 × 4) × (10⁻³ × 10⁸) = 20 × 10⁵. A is too big: 20 = 2 × 10, so the answer is 2 × 10⁶. (b)(i) The ⁻⁰⁵ on the display is just the index −5, so the answer is 4.7 × 10⁻⁵. (ii) A negative index of −5 means dividing 4.7 by 10 five times: 0.000047.

Read the display, then write it properly: the power of 10 in full (× 10⁻⁵, never just ⁻⁵ on the end of A), or the ordinary number with the digits in the right places.

WHAT YOU'VE LEARNED

A quick recap of today's lesson.

The index tells you how big a number is. A just tells you where it sits inside that power of 10.

What you need to know

A number in standard form looks like A × 10ⁿ, where A is at least 1 but less than 10, and n is an integer (positive, negative or zero).
The index n tells you the size of the number; A only places it within that power of 10. Large numbers have positive indices, and numbers between 0 and 1 have negative ones.
If A has slipped out of range, move powers of 10 between A and the index so the value doesn't change.
To multiply or divide, group the A numbers and the powers of 10 separately, and combine the powers with the index laws.
To add or subtract, the powers of 10 are never combined: get both numbers to the same power of 10 (or write them out in full) first.
A calculator does standard-form calculations quickly — but you have to read its display and write the answer properly.

The big picture

Standard form writes a number as A × 10ⁿ, where A is at least 1 but less than 10 and n is an integer. The index n does the heavy lifting: it tells you how big the number is. A just places it within that power of 10. Once that clicks, everything else follows — multiply and divide by grouping the A numbers and the powers, add and subtract by keeping every digit's place value, and always tidy A back between 1 and 10.

Key points

1Standard form: A × 10ⁿ with 1 ≤ A < 10 and n an integer.

2Compare sizes by the index first; compare A only when the indices are equal.

3A too big → divide A by 10, add 1 to the index. A too small → multiply A by 10, subtract 1 from the index.

4Multiply: multiply the A numbers, add the indices, tidy A.

5Divide: write it as a fraction, divide the A numbers, subtract the indices, tidy A.

6Add or subtract: match the powers of 10 first, then combine the A numbers — never the indices.

7A calculator display like 4.7 × 10⁻⁰⁵ means 4.7 × 10⁻⁵, which is 0.000047.

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Worked example

Problem

Write these numbers in order of size, starting with the smallest: 3.2 × 10⁻⁴, 0.0029, 4.1 × 10⁻⁵, 1.05 × 10⁻³

⚠ Watch out

Moving A and the index the same way when tidying up. If A gets 10 times smaller, the power of 10 must get 10 times bigger: 45 × 10³ becomes 4.5 × 10⁴, not 4.5 × 10². With negative indices: 0.7 × 10⁻³ becomes 7 × 10⁻⁴.

🧠

Memory hook

TIMES — the powers team up (add the indices). PLUS — the powers stay put (line up the place values). And whatever you do, finish by tidying A back between 1 and 10.

✓

Check yourself

Work out (8 × 10⁶) ÷ (2 × 10⁻²) and (8 × 10⁶) + (2 × 10⁵) in standard form. Which calculation combined the indices — and why didn't the other?

Flashcards

(14)

What does a number in standard form look like?

A × 10ⁿ, where 1 ≤ A < 10 and n is an integer.

In A × 10ⁿ, which part mainly decides how big the number is?

The index n. A only places the number within that power of 10.

How do you decide which of two standard-form numbers is bigger?

Compare the indices first. Only if the indices are equal do you compare the A numbers.

What does a negative index tell you about a number in standard form?

The number is between 0 and 1 — it is small, not negative. For example, 3 × 10⁻² = 0.03.

Write 52 000 in standard form.

5.2 × 10⁴

Write 0.0061 in standard form.

6.1 × 10⁻³

A is 10 or more. How do you fix it without changing the value?

Divide A by 10 and add 1 to the index (repeat if needed). For example, 23 × 10⁵ = 2.3 × 10⁶.

A is less than 1. How do you fix it without changing the value?

Multiply A by 10 and subtract 1 from the index. For example, 0.8 × 10⁻² = 8 × 10⁻³.

How do you multiply two numbers in standard form?

Group the A numbers and the powers of 10: multiply the A numbers, add the indices, then tidy A.

How do you divide two numbers in standard form?

Write it as a fraction: divide the A numbers, subtract the indices, then tidy A.

When you add two numbers in standard form, do you add the indices?

No. Adding keeps every digit's place value. Match the powers of 10 (or write the numbers in full), add the A numbers, then tidy.

Why is 2.3 × 10⁵ + 4.1 × 10⁵ quick to work out?

The powers of 10 are the same, so you just add the A numbers: 6.4 × 10⁵.

A calculator display shows 6.3 × 10⁻⁰⁴. What is that as an ordinary number?

0.00063 — the ⁻⁰⁴ is the index −4.

Why write very large or very small numbers in standard form?

Long strings of zeros are hard to read and easy to miscount. A × 10ⁿ is short, shows the size at a glance, and makes calculating with them less error-prone.

Tap any card to flip it, or use Study as deck to go through them one at a time. In the full lesson these run as a spaced-repetition deck — you rate each card Hard, Good or Easy and the tricky ones keep coming back until they stick.

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Which is bigger: 9 × 10³ or 1.1 × 10⁴?

Sort each number, then fix the ones that break the rule

You just added indices to multiply. Does that work for adding too?

Adding and subtracting with different powers

One without a calculator, one with

More AQA GCSE Maths topics