Standard Form

Spec reference: Number - Standard form
Key idea: Write very large and very small numbers in standard form and calculate with them.

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What is standard form?

Standard form writes numbers as:

$$A\times {10}^n$$

where $1\le A<10$ and $n$ is an integer.

Number	Standard form
3,400,000	$3.4\times {10}^6$
0.000056	$5.6\times {10}^{-5}$
270	$2.7\times {10}^2$
0.003	$3\times {10}^{-3}$

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Converting to standard form

1. Write the digits with a decimal point after the first non-zero digit
2. Count how many places the decimal point moves - this is $n$
3. Moving right gives a negative power; moving left gives a positive power

Example

Write 45,000 in standard form.

Place decimal after first digit: $4.5$

Decimal moves 4 places left:

$$45000=\boxed{{\displaystyle 4.5\times {10}^4}}$$

Example

Write 0.0000072 in standard form.

Place decimal after first non-zero digit: $7.2$

Decimal moves 6 places right:

$$0.0000072=\boxed{{\displaystyle 7.2\times {10}^{-6}}}$$

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Converting from standard form

Positive power - move decimal point right (large number).
Negative power - move decimal point left (small number).

Example

Write $3.6\times {10}^5$ as an ordinary number.

Move decimal 5 places right: $\boxed{{\displaystyle 360,000}}$

Example

Write $8.1\times {10}^{-4}$ as an ordinary number.

Move decimal 4 places left: $\boxed{{\displaystyle 0.00081}}$

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Adding and subtracting in standard form

Convert to ordinary numbers (or make the powers equal) then add/subtract.

Example

$(3.2\times {10}^4)+(4.5\times {10}^3)$

$=32000+4500=36500=\boxed{{\displaystyle 3.65\times {10}^4}}$

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Multiplying in standard form

Multiply the $A$ values, add the powers.

Example

$(4\times {10}^3)\times (2\times {10}^5)$

$=(4\times 2)\times {10}^{3+5}=8\times {10}^8$

Example

$(3\times {10}^4)\times (5\times {10}^3)$

$=15\times {10}^7=\boxed{{\displaystyle 1.5\times {10}^8}}$ (adjust so $A$ is between 1 and 10)

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Dividing in standard form

Divide the $A$ values, subtract the powers.

Example

$(9\times {10}^7)÷(3\times {10}^2)$

$=(9÷3)\times {10}^{7-2}=\boxed{{\displaystyle 3\times {10}^5}}$

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Exam tips

Watch out for

• After multiplying, check $A$ is still between 1 and 10 - adjust if needed
• ${10}^0=1$, so $5\times {10}^0=5$
• On a calculator, $\times {10}^x$ is entered using the EXP or $\times {10}^x$ button

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Test Yourself

Question 1 of 5

Write 0.000047 in standard form