Standard form writes any number as $A \times 10^{n}$, where $1 \leqslant A < 10$ and $n$ is an integer. It's how calculators and scientists handle very large or very small numbers compactly, and Extended candidates must be able to add, subtract, multiply, and divide values given this way — by hand.

Multiply: multiply the front numbers, add the powers of $10$. Divide: divide the front numbers, subtract the powers of $10$.

Worked Example

Work out $\dfrac{9\times10^{-2}}{2\times10^{5}}$, giving your answer in standard form.

1. Divide front numbers: $9\div2 = 4.5$.
2. Subtract exponents: $(-2)-5 = -7$.
3. Combine: $4.5\times10^{-7}$.

Unlike multiplying and dividing, the powers of $10$ must match before you add or subtract. Rewrite one term so both share the same power, then combine only the front numbers.

Worked Example

Work out $2.6\times10^{5} + 8.4\times10^{4}$, giving your answer in standard form.

1. Rewrite $8.4\times10^4$ with power $10^5$: $8.4\times10^4 = 0.84\times10^5$.
2. Add the front numbers: $(2.6+0.84)\times10^5 = 3.44\times10^5$.

Check the final front number satisfies $1 \leqslant A < 10$. If it doesn't (e.g. you get $32\times10^6$), shift the decimal point and adjust the power to compensate — each place you move the decimal changes the power of $10$ by exactly one: $32\times10^6 = 3.2\times10^7$.

Move the decimal point until exactly one non-zero digit sits before it, and count how many places — and in which direction — you moved it. Moving the point left (for a large number) gives a positive power; moving it right (for a small number, less than $1$) gives a negative power.

Worked Example

Write $0.0000456$ in standard form.

1. Move the decimal point right until one non-zero digit is in front: $4.56$.
2. Count the places moved: $5$ places to the right.
3. Moving right for a small number gives a negative power: $0.0000456 = 4.56\times10^{-5}$.

Calculators typically show standard form using an E or a small raised number, e.g. 4.5E-7 or 4.5⁻⁰⁷, meaning $4.5\times10^{-7}$. Never copy the calculator's shorthand directly into written exam working — always convert it into the full $A\times10^n$ notation, since the shorthand form is not accepted for method or accuracy marks.