Apply the laws of indices for integer, negative and fractional powers; and write, order and calculate with numbers in standard form $a \times 10^n$.

What this dot point is asking

OCR references N7 and N10 combine the laws of indices, including negative and fractional powers at Higher tier, with standard form, the compact way to write very large and very small numbers. The two topics belong together because standard form is index notation in action. Both appear across all three papers, and the index laws are foundational for surds, algebra and growth and decay, so the rules must be automatic.

The laws of indices

The three core laws follow from what a power means: repeated multiplication.

So $3^4 \times 3^2 = 3^6$, $\dfrac{5^7}{5^3} = 5^4$, and $(2^3)^2 = 2^6$. A negative index signals a reciprocal: $4^{-2} = \dfrac{1}{16}$. The laws only combine powers of the same base, so $2^3 \times 3^2$ cannot be simplified to a single power.

Negative and fractional powers (Higher)

Fractional powers fuse roots and powers, and this is a reliable source of Higher-tier marks.

So $27^{2/3} = \left(\sqrt[3]{27}\right)^2 = 3^2 = 9$, and $25^{-1/2} = \dfrac{1}{\sqrt{25}} = \dfrac{1}{5}$. Taking the root before the power keeps the arithmetic small, which matters on the non-calculator paper.

Standard form

Standard form keeps very large and very small numbers manageable.

To order numbers in standard form, compare the powers of $10$ first and only look at the number part if the powers tie. So $3.1 \times 10^{5} > 9.8 \times 10^{4}$ despite $9.8 > 3.1$, because $10^5 > 10^4$.

Calculating in standard form

Multiplication and division use the index laws on the powers of $10$.

For addition and subtraction you cannot simply combine the parts; convert both to ordinary numbers (or to the same power of $10$) first, then add. For instance $3 \times 10^{4} + 5 \times 10^{3} = 30\,000 + 5\,000 = 35\,000 = 3.5 \times 10^{4}$.

Why this matters

Standard form is how science and the calculator display handle the very large (distances in space) and the very small (sizes of cells), and OCR sets contextual questions in exactly these settings. The index laws then reappear in surds, in simplifying algebraic expressions, and in exponential growth and decay. Because a calculator gives standard-form answers automatically, you must be able to read, write and re-standardise them confidently, including spotting when a result like $20 \times 10^{-3}$ needs adjusting to $2 \times 10^{-2}$.