Use the laws of indices for positive, negative and fractional powers, evaluate powers and roots, and write numbers in standard form and calculate with them.

What this dot point is asking

Indices and standard form are the CCEA Number tools for handling powers and for writing very large or very small numbers compactly. You must apply the laws of indices, including negative and fractional powers at Higher tier, evaluate powers and roots, and write numbers in standard form and calculate with them. Standard form appears in science contexts and on the calculator work, while the index laws underpin algebraic manipulation across the Algebra strand.

The laws of indices

An index (or power) tells you how many times to multiply a base by itself. The laws all follow from this.

So $2^3 \times 2^4 = 2^7 = 128$ and $\tfrac{x^8}{x^5} = x^3$. The most common slip is to multiply the indices when the operation is multiplication; the rule is to add the indices when multiplying the powers. The laws only apply when the base is the same, so $2^3 \times 5^2$ cannot be combined into a single power, while $2^3 \times 2^2$ becomes $2^5$. When a coefficient is present, deal with the numbers and the powers separately: $3a^2 \times 4a^5 = 12a^7$, multiplying $3 \times 4$ and adding the indices $2 + 5$. A power outside a bracket applies to everything inside, so $(2x^3)^4 = 2^4 x^{12} = 16x^{12}$, because the $2$ is also raised to the fourth power.

Negative and fractional indices

At Higher tier the powers extend below zero and into fractions.

A negative index means take the reciprocal: $a^{-n} = \dfrac{1}{a^n}$, so $2^{-3} = \tfrac{1}{8}$ and $5^{-1} = \tfrac{1}{5}$. A fractional index means a root: $a^{1/2} = \sqrt{a}$ and, more generally, $a^{1/n} = \sqrt[n]{a}$. Combining the two, $a^{m/n} = (\sqrt[n]{a})^m$, so the denominator is the root and the numerator is the power.

Evaluating powers and roots

To evaluate a power with a fractional index by hand, deal with the root first.

Standard form

Standard form writes a number as $A \times 10^n$, where $1 \le A < 10$ and $n$ is an integer. A positive power makes a large number, so $3.2 \times 10^6 = 3\,200\,000$, while a negative power makes a small number, so $4.5 \times 10^{-3} = 0.0045$. The power of ten records how many places the decimal point has moved away from the position that leaves a single non-zero digit before it. Standard form is valued because it captures both the size and the precision of a quantity in a compact way, which is why scientific data such as the mass of an electron or the distance to a star are written this way.

To multiply or divide in standard form, deal with the number parts and the powers of ten separately, then adjust so the number part lies between 1 and 10. For example, $(6 \times 10^4) \div (2 \times 10^7) = 3 \times 10^{-3}$, dividing $6 \div 2$ and subtracting the powers $4 - 7$. To add or subtract, it is usually easiest to write the numbers out in full first, line up the place values, then convert back, because the powers of ten must match before the number parts can be combined. On a calculator the standard-form key (often labelled $\times 10^x$ or EXP) enters the power directly, and you should read displayed answers such as $1.2\text{E}5$ as $1.2 \times 10^5$.

Why this matters

The index laws are the grammar of algebra, reused in expanding, factorising and rearranging, while standard form is how science and the calculator paper handle quantities from atomic masses to astronomical distances. CCEA tests both the mechanical laws and their application, so fluency here unlocks marks in Number, Algebra and applied contexts alike.