7.5 Standard form — Learn

Prerequisites: multiplying and dividing decimals and using powers of ten.

Before you begin

You should be able to multiply and divide decimals and use powers of ten.

Objectives

• Write numbers in standard form.
• Compare numbers in standard form.
• Multiply and divide numbers in standard form.
• Add or subtract after matching powers of ten.

Writing standard form

Standard form writes a number as $a\times10^n$, where $1\leq a<10$ and $n$ is an integer.

Example 1. Write $5\,630\,000$ in standard form.

The decimal point moved six places left, so the power is $6$.

Example 2. Write $0.000072$ in standard form.

The decimal point moved five places right, so the power is negative.

Calculating in standard form

Multiply the number parts and add the powers. Divide the number parts and subtract the powers.

Example 3. Work out $(3\times10^5)(4\times10^{-2})$.

For addition, first express both terms with the same power.

Example 4. Work out $4.2\times10^6+3\times10^5$.

Watch out. A common error is to leave the first number outside the range $1\leq a<10$. Rewrite $12\times10^3$ as $1.2\times10^4$.

Progress check. Which is greater: $8.1\times10^4$ or $7.9\times10^5$? Compare powers first.

You should now be able to

• write large and small numbers in standard form;
• compare numbers by their powers and first numbers;
• multiply, divide, add and subtract in standard form.