Powers of Ten and Decimals: GCSE Maths Guide

Introduction

Powers of ten sit at the heart of decimals GCSE maths. The core idea is simple: when you multiply or divide a number by a power of 10, every digit shifts to a different place value.

Get this right and topics such as standard form, unit conversions, rounding, and indices become much easier.

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What Are Powers of Ten?

A power of ten is 10 multiplied by itself a set number of times. The exponent tells you how many times 10 is used as a factor.

Negative powers work in reverse and represent fractions or decimals:

• 10⁻¹ = 0.1
• 10⁻² = 0.01
• 10⁻³ = 0.001

Pattern to Remember

For positive powers of 10, the exponent tells you how many zeros follow the 1.

For negative powers of 10, the exponent tells you how far the 1 is positioned after the decimal point.

This is one of the quickest shortcuts in powers of 10 GCSE questions.

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Decimal Place Value

Decimal place value describes the value of each digit according to its position.

Every place-value column is 10 times larger than the column immediately to its right.

Take the number:

47.356

Each column maps directly to a power of ten.

For example, the digit 3 is in the tenths column.

Its place is:

Tenths

Its value is:

0.3

The digit itself is 3, but its value depends on its position.

GCSE Tip

Examiners may ask for either the place of a digit or the value of a digit.

For example, in 4.056:

• The place of the 5 is hundredths
• The value of the 5 is 0.05

These are different answers, so do not confuse them.

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Multiplying Decimals by 10, 100 and 1,000

Rule

When multiplying by powers of 10, the digits move LEFT towards higher place values.

• × 10 → digits shift 1 place left
• × 100 → digits shift 2 places left
• × 1,000 → digits shift 3 places left

Worked Examples Using 3.47

For:

3.47 × 1,000

the digits 3, 4 and 7 all move three place-value columns to the left.

The answer is:

3,470

The final zero is important because it acts as a placeholder.

Sanity Check

Multiplying by a number greater than 1 should make the number larger.

For example:

3.47 × 100 = 347

Since 100 is greater than 1, the answer should be larger than 3.47.

If your answer becomes smaller, you have probably moved the digits in the wrong direction.

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Dividing Decimals by 10, 100 and 1,000

Rule

When dividing by powers of 10, the digits move RIGHT towards lower place values.

• ÷ 10 → digits shift 1 place right
• ÷ 100 → digits shift 2 places right
• ÷ 1,000 → digits shift 3 places right

Worked Examples

For:

56 ÷ 1,000

you can think of 56 as:

56.000

Move the digits three places to the right:

56 → 5.6 → 0.56 → 0.056

Therefore:

56 ÷ 1,000 = 0.056

The placeholder zeros are essential.

Sanity Check

Dividing by a number greater than 1 should make the result smaller.

For example:

56 ÷ 1,000 = 0.056

Since 1,000 is greater than 1, the answer must be smaller than 56.

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Multiplying by 0.1, 0.01 and 0.001

A useful connection is:

0.1 = 1/10

Therefore, multiplying by 0.1 is the same as dividing by 10.

Similarly:

Worked Example 1

84 × 0.1

This is the same as:

84 ÷ 10

So:

84 × 0.1 = 8.4

Worked Example 2

230 × 0.001

This is the same as:

230 ÷ 1,000

Therefore:

230 × 0.001 = 0.23

Important GCSE Point

Multiplying does not always make a number bigger.

When you multiply by a number between 0 and 1, the result becomes smaller.

For example:

50 × 0.1 = 5

Since 0.1 is less than 1, the answer is smaller than 50.

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Common GCSE Mistakes

Moving Digits in the Wrong Direction

Remember:

• Multiply by 10, 100 or 1,000 → digits move left
• Divide by 10, 100 or 1,000 → digits move right

Writing the direction in your working can help prevent errors.

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Forgetting Placeholder Zeros

For example:

3.47 × 1,000 = 3,470

Not:

347

The zero is required to hold the correct place value.

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Assuming Multiplication Always Makes a Number Bigger

This is only true when multiplying by a number greater than 1.

For example:

8 × 10 = 80

But:

8 × 0.1 = 0.8

Multiplying by a decimal less than 1 makes the number smaller.

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Confusing Decimal Places with Significant Figures

These are different ideas.

For example:

0.0560

This number has:

• 4 decimal places
• 3 significant figures

The leading zeros are not significant, but the final zero after the 6 is significant.

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Quick Reference Table

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FAQ

Are Powers of Ten a GCSE Topic?

Yes. Powers of ten appear throughout GCSE Maths and are especially important in the Number topic.

They are also used in:

• Standard form
• Indices
• Decimal calculations
• Unit conversions
• Rounding
• Significant figures

Understanding powers of ten gives you a strong foundation for several other GCSE topics.

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What Is 10⁰?

Any non-zero number raised to the power of 0 equals 1.

Therefore:

10⁰ = 1

In place value, 10⁰ represents the ones column.

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How Do You Multiply a Decimal by 100?

Move every digit two place-value columns to the left.

For example:

4.37 × 100

Shift the digits two places left:

4.37 → 43.7 → 437

Therefore:

4.37 × 100 = 437

Add placeholder zeros if necessary.

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How Do You Divide a Decimal by 100?

Move every digit two place-value columns to the right.

For example:

8.4 ÷ 100

Shift the digits two places right:

8.4 → 0.84 → 0.084

Therefore:

8.4 ÷ 100 = 0.084

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Why Does Multiplying by 0.1 Make a Number Smaller?

Because:

0.1 = 1/10

So multiplying by 0.1 means taking one tenth of the original number.

For example:

70 × 0.1 = 7

This is exactly the same as:

70 ÷ 10 = 7

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Summary

The rules are consistent once you understand the place-value pattern:

• × 10 → digits shift left 1 place
• × 100 → digits shift left 2 places
• × 1,000 → digits shift left 3 places
• ÷ 10 → digits shift right 1 place
• ÷ 100 → digits shift right 2 places
• ÷ 1,000 → digits shift right 3 places
• × 0.1 → same as ÷ 10
• × 0.01 → same as ÷ 100
• × 0.001 → same as ÷ 1,000
• Always use placeholder zeros where necessary
• Multiplying by a number greater than 1 makes the value larger
• Multiplying by a number between 0 and 1 makes the value smaller

Mastering these rules gives you the foundation for standard form, indices, decimal calculations, rounding, and unit conversions in GCSE Maths.