Number - Structure and calculation: calculate with and interpret standard form A × 10ⁿ, where 1 ≤ A < 10 and n is an integer

The idea

This notation writes any number as a value that is at least 1 and below 10, multiplied by a power of ten, which keeps very large and very small numbers readable. So 4300 becomes 4.3 × 10³, and 0.00062 becomes 6.2 × 10⁻⁴, where the negative index says a positive number smaller than 1. To multiply, multiply the front numbers and add the indices: (3 × 10⁵) × (2 × 10⁴) = 6 × 10⁹. To divide, divide the front numbers and subtract the indices. To add or subtract, rewrite both with the same power of ten first, or turn both into ordinary numbers and convert back at the end. Tidy the answer so the front number lands back in range, since 12 × 10⁶ is not yet in this form and should be written as 1.2 × 10⁷.

What earns the marks

• A number converted correctly, with the front number at least 1 and below 10.
• Adding the indices when multiplying, or subtracting them when dividing.
• A correct but untidied answer such as 0.4 × 10⁵ before it is adjusted.
• The final answer in standard form, which needs both the front number and the index right.

Common mistakes and how to put them right

• Leaving the front number outside the range, so offering 0.45 × 10⁴ as a final answer.

• Counting digits rather than place value shifts, so writing 0.0035 as 3.5 × 10⁻² rather than 3.5 × 10⁻³.

• Adding the front numbers as they stand when the two powers of ten are different.

• Copying a calculator display in its own shorthand instead of rewriting it in standard form.

Exam tips

• Check the index against the size of the number: a positive number smaller than 1 always takes a negative index.
• When adding or subtracting and the indices are close, converting to ordinary numbers is often faster than matching the powers.
• Give the answer in standard form whenever the question asks for it, even when the numbers you were given were written out in full.