Apply the laws of indices (including zero, negative and fractional indices at Higher tier); and write numbers in standard form and calculate with them, both with and without a calculator.

What this dot point is asking

The Eduqas number content requires fluency with the laws of indices and with standard form. You must simplify expressions using the index laws, evaluate zero, negative and fractional indices (Higher tier), write very large and very small numbers in standard form, and calculate with standard-form numbers both with and without a calculator. Standard form is the language of science, so it appears in cross-topic and contextual questions, and the index laws underpin algebraic manipulation later in the course. Both Eduqas components test this content.

The laws of indices

The index laws turn repeated multiplication into simple arithmetic on the powers.

So $x^5 \times x^3 = x^8$, $\dfrac{y^7}{y^2} = y^5$, and $(z^4)^3 = z^{12}$. The laws only combine powers of the same base, so $2^3 \times 3^2$ cannot be simplified to a single power. At Higher tier the family extends to negative and fractional indices: $a^{-n} = \tfrac{1}{a^n}$ turns a negative index into a reciprocal, and $a^{m/n} = \left(\sqrt[n]{a}\right)^m$ links indices to roots, so $8^{2/3} = \left(\sqrt[3]{8}\right)^2 = 2^2 = 4$.

Evaluating zero, negative and fractional indices (Higher)

These three special cases are reliable Higher marks.

Taking the negative sign first (flip), then the root, then the power, keeps the numbers small and the working clear.

Standard form

Standard form expresses numbers as a manageable digit string times a power of ten.

To convert $0.000\,072$ to standard form, move the point so one non-zero digit sits in front, giving $7.2$, and count the places moved, here five to the right, giving $7.2 \times 10^{-5}$. To convert back, move the point the other way by the size of the index.

Calculating with standard form

The rules mirror the index laws applied to the powers of ten.

To multiply or divide standard-form numbers, combine the number parts and add or subtract the powers: $(4 \times 10^5) \times (2 \times 10^3) = 8 \times 10^8$, and $(6 \times 10^7) \div (3 \times 10^2) = 2 \times 10^5$. If the resulting number part falls outside $1$ to $10$, re-standardise: $(5 \times 10^6) \times (4 \times 10^3) = 20 \times 10^9 = 2 \times 10^{10}$. To add or subtract, first write both numbers to the same power of ten (or as ordinary numbers), then combine.

Why this matters

Standard form is how the Eduqas exam handles the astronomically large and the microscopically small, in physics, chemistry and finance contexts, and the index laws are the grammar of algebra: simplifying $\frac{6x^5}{2x^2}$ or expanding $(2y^3)^2$ uses exactly these rules. Because Component 1 is non-calculator, the index evaluations and standard-form multiplications must be done by hand, while Component 2 expects fluent use of the calculator's standard-form button.