Year 9 Scientific Notation Worksheets

Worked examples

Year 9 scientific notation: worked examples

Three examples, from a standard question to a worded problem. Have a go at each one before you open the solution.

1. Example 1 · Converting between scientific notation and decimal form

   Write each number in the form requested.

   (a) Write $3.07 \times 10^5$ as an ordinary number.

   (b) Write $0.000 \, 048$ in scientific notation.

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   Key idea: In scientific notation a number is written as $a \times 10^n$, where $1 \leq a < 10$ and $n$ is an integer. A positive power of 10 makes the number larger; a negative power makes it smaller.

   1. (a) $3.07 \times 10^5$ means move the decimal point 5 places to the right: $3.07 \times 10^5 = 307\,000$
   2. (b) The first significant digit in $0.000\,048$ is 4. Count how many places the decimal point must move right to give a number between 1 and 10: it moves 5 places, giving $4.8$. Because the original number is less than 1, the power is negative: $0.000\,048 = 4.8 \times 10^{-5}$

   Answer: (a) $307\,000$ and (b) $4.8 \times 10^{-5}$

2. Example 2 · Multiplying and dividing in scientific notation

   Evaluate each of the following, giving your answer in scientific notation.

   (a) $(3.2 \times 10^4) \times (4.5 \times 10^6)$

   (b) $\dfrac{6.6 \times 10^3}{2.4 \times 10^{-2}}$

   Show worked solution

   Key idea: Multiply (or divide) the coefficients separately from the powers of 10, using index laws $10^m \times 10^n = 10^{m+n}$ and $10^m \div 10^n = 10^{m-n}$. Adjust if the coefficient falls outside $[1, 10)$.

   1. (a) Multiply the coefficients and add the indices:
      $\begin{aligned} (3.2 \times 4.5) \times 10^{4+6} &= 14.4 \times 10^{10} \end{aligned}$
   2. $14.4$ is not in $[1, 10)$, so adjust: $14.4 = 1.44 \times 10^1$, giving $1.44 \times 10^{11}$
   3. (b) Divide the coefficients and subtract the indices:
      $\begin{aligned} \frac{6.6}{2.4} \times 10^{3-(-2)} &= 2.75 \times 10^{5} \end{aligned}$
   4. $2.75$ is already in $[1, 10)$, so no adjustment is needed.

   Answer: (a) $1.44 \times 10^{11}$ and (b) $2.75 \times 10^{5}$

3. Example 3 · Real-world problem: comparing distances in space

   The distance from Earth to the Sun is approximately $1.496 \times 10^{11}$ metres. The distance from Earth to the Moon is approximately $3.84 \times 10^{8}$ metres. How many times farther from Earth is the Sun than the Moon? Give your answer correct to the nearest whole number.

   Show worked solution

   Key idea: "How many times farther" means dividing the larger distance by the smaller distance. Divide numbers in scientific notation by handling the coefficients and powers of 10 separately.

   1. Set up the division:
      $\frac{1.496 \times 10^{11}}{3.84 \times 10^{8}}$
   2. Divide the coefficients:
      $\frac{1.496}{3.84} \approx 0.38958\ldots$
   3. Divide the powers of 10:
      $10^{11} \div 10^{8} = 10^{3}$
   4. Combine and simplify:
      $0.38958\ldots \times 10^{3} = 389.58\ldots$
   5. Round to the nearest whole number: $\approx 390$.

   Answer: The Sun is approximately $390$ times farther from Earth than the Moon.

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