Standard Form

This lesson covers standard form (also called standard index form or scientific notation) for AQA GCSE Mathematics. Standard form is used to write very large or very small numbers in a compact and manageable way. You need to be able to convert numbers into and out of standard form, and perform calculations with numbers in standard form.

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What Is Standard Form?

A number in standard form is written as:

a x 10 to the power n

where:

• a is a number between 1 and 10 (1 <= a < 10)
• n is an integer (positive or negative)

Examples

Ordinary Number	Standard Form
4,500	4.5 x 10 cubed
72,000,000	7.2 x 10 to the power 7
340	3.4 x 10 squared
0.006	6 x 10 to the power (-3)
0.0000045	4.5 x 10 to the power (-6)
9,100,000	9.1 x 10 to the power 6

Exam Tip: The value of "a" must be at least 1 and less than 10. A common mistake is writing 45 x 10 squared instead of 4.5 x 10 cubed. Both equal 4,500, but only the second is correct standard form.

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Converting to Standard Form

Large Numbers (n is positive)

1. Place the decimal point after the first non-zero digit to find "a".
2. Count how many places the decimal point has moved. This gives "n".

Worked Example

Write 6,340,000 in standard form.

Step 1: a = 6.34 (decimal after the first digit)

Step 2: The decimal point moved 6 places to the left.

Answer: 6.34 x 10 to the power 6

Small Numbers (n is negative)

1. Place the decimal point after the first non-zero digit to find "a".
2. Count how many places the decimal point has moved. Use a negative power.

Worked Example

Write 0.000372 in standard form.

Step 1: a = 3.72

Step 2: The decimal point moved 4 places to the right.

Answer: 3.72 x 10 to the power (-4)

graph TD
    A["Number to Standard Form"] --> B{"Is the number >= 1?"}
    B -->|Yes large number| C["Move decimal left"]
    B -->|No small number| D["Move decimal right"]
    C --> E["Power of 10 is POSITIVE"]
    D --> F["Power of 10 is NEGATIVE"]
    E --> G["a x 10 to the power n"]
    F --> G

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Converting from Standard Form

Reverse the process: move the decimal point according to the power of 10.

• Positive power: move the decimal point to the right (number gets bigger).
• Negative power: move the decimal point to the left (number gets smaller).

Worked Examples

Standard Form	Direction	Ordinary Number
3.7 x 10 to the power 5	Right 5 places	370,000
8.04 x 10 cubed	Right 3 places	8,040
2.5 x 10 to the power (-3)	Left 3 places	0.0025
6.1 x 10 to the power (-5)	Left 5 places	0.000061

Exam Tip: A quick check — if the power is positive, the ordinary number should be large. If the power is negative, the ordinary number should be small (less than 1).

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Ordering Numbers in Standard Form

To order numbers in standard form:

1. Compare the powers of 10 first — a larger power means a larger number (for positive numbers).
2. If the powers are the same, compare the values of "a".

Worked Example

Order these from smallest to largest: 3.2 x 10 to the power 4, 7.1 x 10 cubed, 5.6 x 10 to the power 4, 1.9 x 10 to the power 5

Value	Power	a
7.1 x 10 cubed	3	7.1
3.2 x 10 to the power 4	4	3.2
5.6 x 10 to the power 4	4	5.6
1.9 x 10 to the power 5	5	1.9

Answer: 7.1 x 10 cubed, 3.2 x 10 to the power 4, 5.6 x 10 to the power 4, 1.9 x 10 to the power 5

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Calculations with Standard Form

Multiplying

Multiply the "a" values and add the powers.

Calculate (3 x 10 to the power 4) x (5 x 10 cubed)

Step 1: 3 x 5 = 15

Step 2: 10 to the power 4 x 10 cubed = 10 to the power 7

Step 3: 15 x 10 to the power 7

Step 4: Adjust to standard form: 15 = 1.5 x 10, so 1.5 x 10 x 10 to the power 7 = 1.5 x 10 to the power 8

Dividing

Divide the "a" values and subtract the powers.

Calculate (8.4 x 10 to the power 6) / (2.1 x 10 squared)

Step 1: 8.4 / 2.1 = 4

Step 2: 10 to the power 6 / 10 squared = 10 to the power 4

Answer: 4 x 10 to the power 4

Exam Tip: After multiplying or dividing, always check that your answer is in correct standard form (1 <= a < 10). If "a" is 10 or more, adjust by increasing the power. If "a" is less than 1, adjust by decreasing the power.

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Adding and Subtracting in Standard Form

To add or subtract, the numbers must have the same power of 10. If they do not, convert to ordinary numbers or adjust the power.

Worked Example

Calculate (4.5 x 10 to the power 5) + (3.2 x 10 to the power 4)

Method 1 — Convert to ordinary numbers:

• 4.5 x 10 to the power 5 = 450,000
• 3.2 x 10 to the power 4 = 32,000
• 450,000 + 32,000 = 482,000

Answer: 4.82 x 10 to the power 5

Method 2 — Make the powers the same:

• 3.2 x 10 to the power 4 = 0.32 x 10 to the power 5
• (4.5 + 0.32) x 10 to the power 5 = 4.82 x 10 to the power 5

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