Mastering Standard Form for Your Mathematics Exam
This guide cuts through the noise to explain standard form directly. Learn the method, see worked examples, avoid common mistakes, and practice with exam-style questions.
What the examiner is testing
The examiner is assessing your ability to convert between ordinary numbers and standard form, and to perform calculations with numbers in standard form. Marks are awarded for correctly identifying the power of 10 and the number between 1 and 10.
The method
- Identify the significant digits: Locate the first non-zero digit in the number.
- Form the 'a' part: Create a new number by placing a decimal point immediately after the first non-zero digit. This number, \(a\), must satisfy \(1 \le a < 10\).
- Count decimal place shifts: Count how many places the original decimal point (or implied decimal point for whole numbers) had to move to reach its new position in step 2.
- Determine the power of 10: This count is the power of 10, \(n\). If the original number was large (greater than or equal to 10), \(n\) is positive. If the original number was small (between 0 and 1), \(n\) is negative.
- Combine into standard form: Write the number as \(a \times 10^n\).
Worked example
Convert \(56,700,000\) to standard form.
$$
56,700,000
$$
1. Significant digits: \(567\)
2. Form 'a' part: \(5.67\)
3. Count decimal shifts: The original decimal point is at the end of \(56,700,000\). To get \(5.67\), it moved 7 places to the left.
4. Determine power of 10: Since \(56,700,000\) is a large number, the power is positive. So, \(n = 7\).
5. Combine: \(5.67 \times 10^7\)
Sanity check: \(5.67 \times 10^7 = 5.67 \times 10,000,000 = 56,700,000\). The answer matches the original number.
Worked example: a harder one
Calculate \((4.2 \times 10^5) \times (2 \times 10^{-3})\). Give your answer in standard form.
A common first move is to multiply the numbers and powers separately:
$$
(4.2 \times 2) \times (10^5 \times 10^{-3})
$$
$$
8.4 \times 10^{5 + (-3)}
$$
$$
8.4 \times 10^2
$$
This is already in standard form, so no further adjustment is needed. This 'obvious' move works here.
However, consider if the numbers were different: Calculate \((7 \times 10^4) \times (6 \times 10^2)\).
The 'obvious' first move:
$$
(7 \times 6) \times (10^4 \times 10^2)
$$
$$
42 \times 10^{4+2}
$$
$$
42 \times 10^6
$$
This is NOT in standard form because \(42\) is not between 1 and 10. This is why the 'obvious' move fails if you stop here.
To correct this:
1. Convert \(42\) to standard form: \(4.2 \times 10^1\).
2. Substitute this back into the expression:
$$
(4.2 \times 10^1) \times 10^6
$$
3. Combine the powers of 10:
$$
4.2 \times 10^{1+6}
$$
$$
4.2 \times 10^7
$$
This is the correct answer in standard form.
Practice
- Write \(0.00000305\) in standard form.
- Write \(7.81 \times 10^6\) as an ordinary number.
- Calculate \((1.5 \times 10^4) + (3 \times 10^3)\). Give your answer in standard form.
- The diameter of a red blood cell is approximately \(7.5 \times 10^{-6}\) metres. The diameter of a human hair is approximately \(8 \times 10^{-5}\) metres. How many times larger is the diameter of a human hair compared to a red blood cell? Give your answer to 1 decimal place.
Answers:
- \(3.05 \times 10^{-6}\)
- \(7,810,000\)
- \(1.8 \times 10^4\)
- Working for question 4:
To find out how many times larger, we divide the larger value by the smaller value.
$$ \frac{8 \times 10^{-5}}{7.5 \times 10^{-6}} $$
Separate the numbers and the powers of 10:
$$ \left(\frac{8}{7.5}\right) \times \left(\frac{10^{-5}}{10^{-6}}\right) $$
Calculate the numerical part:
$$ \frac{8}{7.5} \approx 1.0666... $$
Calculate the powers of 10 part using index laws (\(a^m / a^n = a^{m-n}\)):
$$ 10^{-5 - (-6)} = 10^{-5 + 6} = 10^1 $$
Combine these results:
$$ 1.0666... \times 10^1 $$
$$ 10.666... $$
Rounding to 1 decimal place:
$$ 10.7 $$
So, the human hair is approximately 10.7 times larger.
The three mistakes that lose marks
- Incorrect 'a' value: The number before the \(\times 10\) must be between 1 and 10 (inclusive of 1, exclusive of 10).
- Wrong answer: \(45 \times 10^3\) (for \(45,000\)). This is not standard form because \(45 \ge 10\).
- Incorrect power of 10: Miscounting the decimal places or getting the sign wrong.
- Wrong answer: \(3.2 \times 10^4\) (for \(0.00032\)). This is incorrect; the power should be negative for small numbers.
- Not giving the final answer in standard form: After calculations, the result might not be in standard form and needs adjustment.
- Wrong answer: \(12.5 \times 10^7\) (after calculating \((5 \times 10^3) \times (2.5 \times 10^4)\)). This needs to be adjusted to \(1.25 \times 10^8\).
30-second recap
Standard form expresses numbers as \(a \times 10^n\), where \(1 \le a < 10\) and \(n\) is an integer. The sign of \(n\) depends on whether the original number was large (\(n\) positive) or small (\(n\) negative). Remember to adjust your final answer to standard form after calculations.