Apply the laws of indices including zero, negative and fractional powers, and express and calculate with numbers in standard form

What this dot point is asking

SEAB wants you to apply the laws of indices, including zero, negative and fractional powers, and to write and compute with numbers in standard form (scientific notation). These tools let you handle powers compactly and deal with very large and very small quantities without long strings of zeros.

The answer

The laws of indices

For the same base, the index laws are:

Add indices when multiplying, subtract when dividing, and multiply when raising a power to a power. A power of a product distributes: $(ab)^n = a^n b^n$.

Zero and negative indices

Any non-zero number to the power zero is $1$, so $7^0 = 1$. A negative index is a reciprocal:

so $2^{-3} = \dfrac{1}{8}$.

Fractional indices

The denominator of a fractional index is a root and the numerator is a power:

For example $8^{2/3} = \left(\sqrt[3]{8}\right)^2 = 2^2 = 4$.

Standard form

Standard form writes a number as $A \times 10^n$, where $1 \le A < 10$ and $n$ is an integer. Large numbers have positive $n$; small numbers have negative $n$. So $4\,500\,000 = 4.5 \times 10^6$ and $0.00072 = 7.2 \times 10^{-4}$.

Calculating in standard form

Multiply or divide the leading numbers and add or subtract the powers of ten, then adjust so the leading number lies between $1$ and $10$:

Adding and subtracting in standard form

Multiplication and division in standard form are straightforward, but addition and subtraction need the powers of ten to match first. Rewrite the numbers so they share the same power of ten, then add or subtract the leading parts, and finally adjust back to proper standard form. To compute $3 \times 10^4 + 5 \times 10^3$, rewrite the second as $0.5 \times 10^4$, so the sum is $3.5 \times 10^4$. The step students skip is aligning the powers, which is essential because $10^4$ and $10^3$ are different units of size that cannot be combined directly.

Comparing numbers in standard form

Standard form makes comparing very large or very small numbers quick: compare the powers of ten first, and only if those are equal compare the leading numbers. So $4 \times 10^6$ is larger than $9 \times 10^5$ despite the smaller leading digit, because $10^6 > 10^5$. For negative powers (small numbers), a less negative power is larger, so $2 \times 10^{-3}$ exceeds $8 \times 10^{-5}$. Ordering a list of numbers by their power of ten first, then by leading digit, is the reliable method and a frequent E-Maths task that catches out anyone who only looks at the leading number.

Examples in context

Example 1. Distances in astronomy. The distance from the Earth to the Sun is about $1.5 \times 10^{11}\ \text{m}$. Standard form lets scientists write and compare such huge distances without dozens of zeros, and arithmetic on them reduces to handling the leading numbers and the powers of ten.

Example 2. Sizes in biology. A bacterium might be $2 \times 10^{-6}\ \text{m}$ across. Negative powers of ten make tiny measurements manageable, and dividing two such measurements quickly gives a ratio of sizes.

Try this

Q1. Evaluate $5^{0} + 3^{-2}$. [2 marks]

• Cue. $5^0 = 1$ and $3^{-2} = \dfrac{1}{9}$, so the sum is $1\dfrac{1}{9}$ or $\dfrac{10}{9}$.

Q2. Write $0.000\,036$ in standard form. [1 mark]

• Cue. Move the point so the leading digit is between $1$ and $10$: $3.6 \times 10^{-5}$.

Q3. Evaluate $16^{3/4}$. [2 marks]

• Cue. Fourth root then cube: $\left(\sqrt[4]{16}\right)^3 = 2^3 = 8$.