Standard form (or scientific notation) is a crucial part of the GCSE Maths curriculum, appearing on all major UK exam boards like AQA, Edexcel, and OCR. It's a way to write very large or very small numbers concisely, using powers of ten. While multiplying and dividing in standard form is often seen as straightforward, adding and subtracting can trip up students if they don't follow a clear method.
This guide will walk you through two reliable methods, complete with a worked example. By the end, you'll be confident in tackling these questions in your exams.
Why is Adding/Subtracting Standard Form Different?
A number in standard form is written as a × 10n, where 1 ≤ a < 10 and n is an integer. You cannot simply add the a values and the n values separately. For example:
(3 × 104) + (2 × 103) ≠ 5 × 107
The exponents (the n in 10n) must be the same before you can combine the numbers. This leads us to our two main strategies.
Method 1: Matching the Exponents (Recommended)
This is the most efficient method and the one you should aim to use in exams. It keeps your working in standard form.
Step-by-Step Process
flowchart TD
A["Start: Identify the two numbers, in standard form"] --> B["Are the exponents, the same?"]
B -- Yes --> C["Add/Subtract the a values, Keep the exponent the same"]
B -- No --> D["Adjust the number with the, smaller exponent"]
D --> E["Increase its exponent to match, the larger one"]
E --> F["This means you must DECREASE, the a value by moving the decimal point"]
F --> C
C --> G["Check your answer is in, correct standard form 1 ≤ a < 10"]
G --> H["Finished"]
Key Rule: To increase the exponent by 1, you must ÷ the a value by 10 (move the decimal point one place left). To decrease the exponent by 1, multiply a by 10 (move the decimal point one place right).
Worked Example
Calculate (4.2 × 105) + (3.1 × 104).
Identify the exponents: They are 5 and 4. They are not the same.
Adjust the number with the smaller exponent (104): We need to change 3.1 × 104 so it has an exponent of 5.
To change 104 to 105, we increase the exponent by 1.
Therefore, we must decrease the a value (3.1) by a factor of 10.
3.1 ÷ 10 = 0.31.
So, 3.1 × 104 = 0.31 × 105.
Now add: (4.2 × 105) + (0.31 × 105)
Add the a values: 4.2 + 0.31 = 4.51
Keep the exponent: 4.51 × 105
Check standard form: 4.51 is between 1 and 10, so this is correct.
Final Answer: 4.51 × 105
Method 2: Converting to Ordinary Numbers
This method involves converting both numbers back to their ordinary (decimal) form, performing the calculation, and then converting the answer back to standard form. It can be helpful for visual learners or for checking your work, but it's prone to errors with very large or very small numbers where counting decimal places is tricky.
Step-by-Step Process
Convert each number from standard form to an ordinary number.
Perform the addition or subtraction with these ordinary numbers.
Convert the result back into correct standard form.
Same Worked Example
Calculate (4.2 × 105) + (3.1 × 104).
Convert:
4.2 × 105 = 420,000 (move decimal point 5 places right).
3.1 × 104 = 31,000 (move decimal point 4 places right).
Add: 420,000 + 31,000 = 451,000
Convert back: 451,000 = 4.51 × 105 (decimal point moved 5 places left).
Final Answer: 4.51 × 105
Practice Problem: Subtraction
Let's try a subtraction using Method 1 (matching exponents).
Calculate (6.8 × 10−3) − (9.2 × 10−4). Write your answer in standard form.
Walkthrough:
Exponents are −3 and −4. Not the same.
Adjust the number with the smaller exponent (10−4). We need 10−3.
To change 10−4 to 10−3, increase exponent by 1.
Therefore, decrease a by 10: 9.2 ÷ 10 = 0.92.
So, 9.2 × 10−4 = 0.92 × 10−3.
Now subtract: (6.8 × 10−3) − (0.92 × 10−3)
Subtract the a values: 6.8 − 0.92 = 5.88
Keep the exponent: 5.88 × 10−3
This is already in correct standard form.
Answer: 5.88 × 10−3
Top Tips and Common Pitfalls
Always make the exponents the same before adding/subtracting the a values. This is the golden rule.
Adjust the number with the smaller exponent. It's easier to make the smaller exponent larger to match the bigger one.
Check your final answer is in standard form. Ensure 1 ≤ a < 10. If not, adjust it.
Example: 15.2 × 106 is not in standard form. Correct it to 1.52 × 107 (increase exponent by 1, decrease a by 10).
Be extra careful with negative exponents. The principle is exactly the same. 10−4 is smaller than 10−3 because −4 < −3.
For subtraction, your final a value might be less than 1. If so, you'll need to convert it back.
Example: Result is 0.76 × 105. Correct it to 7.6 × 104 (decrease exponent by 1, increase a by 10).
Summary
Adding and subtracting with standard form requires one key step: matching the powers of 10. Whether you choose to adjust the exponents directly or convert to ordinary numbers first, the core idea remains. For your GCSE exam, practising the matching exponents method will make you faster and more accurate. Remember to always present your final answer in correct standard form—examiners will check for this!
Ready to test yourself? Try this final challenge:
Calculate (5.6 × 108) + (4.73 × 109). Give your answer in standard form.
(Hint: Which exponent is larger? Adjust the other number to match it.)