Apply the laws of indices including zero, negative and fractional powers (Higher tier), and write and calculate with numbers in standard form.

What this dot point is asking

WJEC requires you to apply the laws of indices, including the zero, negative and fractional powers met at Higher tier, and to write and calculate with numbers in standard form. Standard form is how the specification handles very large and very small quantities, and the index laws underpin algebraic manipulation and surds, so this topic feeds directly into both number and algebra. It appears across both components, with the harder fractional and negative powers reserved for Higher.

The laws of indices

Indices are repeated multiplication, and the laws all follow from that.

So $x^4 \times x^3 = x^7$, $\tfrac{y^9}{y^2} = y^7$ and $(z^2)^5 = z^{10}$. These laws only apply when the base is the same.

Negative and fractional powers (Higher)

At Higher tier the powers extend beyond positive whole numbers.

It is almost always easier to take the root first to keep the numbers small. So $27^{2/3}$ is best done as $(\sqrt[3]{27})^2 = 3^2 = 9$, and a combined case like $16^{-1/2} = \tfrac{1}{16^{1/2}} = \tfrac{1}{4}$ applies both rules: the negative gives the reciprocal and the half gives the square root.

Standard form

Standard form expresses any number compactly as a single digit, a point, then a power of ten.

To add or subtract numbers in standard form, convert them to ordinary numbers (or to a common power of ten) first, then combine and re-standardise the result.

Converting between standard and ordinary form

Reading standard form back into an ordinary number is a frequent first step. For $3.6 \times 10^5$, the positive power $5$ means the number is large, so move the point five places right to get $360\,000$. For $7.2 \times 10^{-3}$, the negative power means it is small, so move three places left to get $0.0072$. Going the other way, count how many places the point must move to leave a single non-zero digit in front, and that count (with the correct sign) is the power. Practising both directions stops the sign of the power being guessed.

Combining the index laws

Harder questions chain several laws together. To simplify $\dfrac{(2^3)^2 \times 2^4}{2^5}$, deal with the bracket first: $(2^3)^2 = 2^6$. The top becomes $2^6 \times 2^4 = 2^{10}$, and dividing by $2^5$ gives $2^{10 - 5} = 2^5 = 32$. The order matches BIDMAS: powers of powers, then multiplication (add), then division (subtract). Keeping the base the same throughout is essential, because the laws only apply to a common base, and an answer is often expected as a single power rather than a fully evaluated number.

Why this matters

The index laws are not just a number topic: they reappear constantly in algebra when simplifying expressions like $\tfrac{6x^5}{2x^2}$, and the same fractional-power thinking underlies surds. Standard form lets the specification pose realistic questions about astronomical distances or atomic sizes without unwieldy strings of zeros. Because WJEC tests the index laws on the non-calculator Unit 1, knowing them by heart, especially the negative and fractional rules, secures quick, reliable marks.