The laws of indices including zero, negative and fractional powers, and standard form: writing very large and very small numbers and calculating with them.

What this dot point is asking

Edexcel expects you to apply the laws of indices confidently, including zero, negative and fractional powers, and to use standard form to write and calculate with very large and very small numbers. Indices underpin standard form, surds and algebra, so a firm grasp here pays off repeatedly. Standard form appears on both the non-calculator and calculator papers.

The laws of indices

The index laws let you simplify powers of the same base without writing them out in full.

So $x^4 \times x^3 = x^7$, $\dfrac{y^9}{y^2} = y^7$, and $(z^2)^5 = z^{10}$. These rules only work when the base is the same, so $2^3 \times 5^2$ cannot be simplified this way.

Zero, negative and fractional powers

Three special cases extend the rules to all rational powers.

• Zero power: any non-zero number to the power $0$ is $1$. So $7^0 = 1$ and $(2x)^0 = 1$. This follows from $a^n \div a^n = a^0 = 1$.
• Negative power: a negative power is the reciprocal. $a^{-n} = \dfrac{1}{a^n}$, so $2^{-3} = \dfrac{1}{8}$ and $\left(\dfrac{3}{4}\right)^{-1} = \dfrac{4}{3}$.
• Fractional power: the denominator is a root and the numerator is a power. $a^{1/2} = \sqrt{a}$, $a^{1/3} = \sqrt[3]{a}$, and $a^{m/n} = \left(\sqrt[n]{a}\right)^m$.

Writing numbers in standard form

Standard form expresses a number as a single non-zero digit, optional decimals, multiplied by a power of ten.

For $4\,500\,000$, move the decimal point so the first number is between $1$ and $10$: $4.5$, and the point moved $6$ places, so it is $4.5 \times 10^6$. For $0.00072$, the first significant digit is $7$, giving $7.2$, and the point moves $4$ places the other way, so $7.2 \times 10^{-4}$.

Calculating with standard form

To multiply or divide, handle the number parts and the powers of ten separately, using the index laws for the powers. For $(3 \times 10^4) \times (2 \times 10^5)$: numbers $3 \times 2 = 6$, powers $10^{4+5} = 10^9$, giving $6 \times 10^9$. Always check the final answer is in valid standard form, adjusting if the number part falls outside the range $1$ to $10$. For example $6 \times 10^3 \times 4 \times 10^3 = 24 \times 10^6$, which must be rewritten as $2.4 \times 10^7$.

Adding and subtracting in standard form is trickier, because the powers of ten must match first. To work out $3.2 \times 10^5 + 4 \times 10^4$, rewrite the second term with the same power: $4 \times 10^4 = 0.4 \times 10^5$. Now add the number parts: $3.2 + 0.4 = 3.6$, giving $3.6 \times 10^5$. On the calculator papers you can type standard form directly using the $\times 10^x$ (or EXP) button, which avoids these adjustments, but you must still write the final answer in correct standard form because the calculator may display it differently.

Why standard form is used

Standard form exists to make extreme numbers manageable. The distance to the Sun is about $1.5 \times 10^{11}\,\text{m}$, and the mass of an electron is about $9.1 \times 10^{-31}\,\text{kg}$; writing these in full would be error-prone and hard to compare. Standard form also makes the size of a number obvious at a glance from the power of ten, so $7 \times 10^9$ is instantly seen to be a thousand times bigger than $7 \times 10^6$. This is why science questions and large-data contexts in the exam almost always use it.