Mathematics • Year 9 • Unit 1 • Lesson 17
Mixed Operations in Scientific Notation
Choose the right rule for each problem: $\times$ adds indices, $\div$ subtracts indices, and $+ / -$ need matching powers first. Re-standardise into $[1, 10)$ at every step. Then debug another student's working and tackle an open-ended composition challenge.
1. Mixed problems — choose the right rule
Each question uses a different combination of operations from Lesson 17. Decide whether to add or subtract indices, or match powers first, before you start writing. Show your working. 3 marks each
1.1 Calculate $(5 \times 10^{-2})(3 \times 10^6)$, giving the answer in scientific notation.
1.2 Calculate $\dfrac{8 \times 10^6}{2 \times 10^2}$, then subtract $1 \times 10^3$ from your answer. Give the final answer in scientific notation.
1.3 Calculate $6 \times 10^{-2} - 5 \times 10^{-3}$ in scientific notation.
1.4 A textbook lists the answer to $(2.5 \times 10^4) \times (4 \times 10^{-7})$ as $1 \times 10^{-2}$. Verify the answer by showing every step, including the re-standardise.
1.5 Find $n$ if $(4 \times 10^n)(2 \times 10^3) = 8 \times 10^7$, and check your answer by substituting back in.
1.6 Calculate $\dfrac{(6 \times 10^5)(5 \times 10^{-2})}{3 \times 10^2}$ fully, in scientific notation.
2. Find the mistake
Another student has tried to calculate $4 \times 10^5 + 3 \times 10^4$. Their working is shown below. Exactly one line contains a mistake. Spot it, explain why it's wrong, then re-do the working correctly. 3 marks
Student's working — calculate $4 \times 10^5 + 3 \times 10^4$:
Line 1: Common power $\to$ rewrite $3 \times 10^4 = 0.3 \times 10^5$.
Line 2: Add coefficients: $4 + 0.3 = 4.3$.
Line 3: Add the powers as well: $10^5 + 10^5 = 10^{10}$.
Line 4: So $4 \times 10^5 + 3 \times 10^4 = 4.3 \times 10^{10}$.
(a) Which line contains the mistake?
(b) Explain in one or two sentences why that line is wrong.
(c) Write out the corrected working in full, including the corrected final answer.
Stuck? Once the powers of 10 are made to match in Step 1, the $10^5$ doesn't get added or multiplied — it just stays as the common factor while the coefficients combine.3. Open-ended challenge — build the answer
This question has more than one valid answer — there are several different ways to build the target. 4 marks
3.1 Find three different expressions in scientific notation — each combining at least one $\times$, $\div$, $+$ or $-$ operation — that all simplify to $\mathbf{6 \times 10^7}$.
For each expression you find:
(i) Write it down.
(ii) Show the working that confirms it equals $6 \times 10^7$.
(iii) Briefly state which operation(s) and rule(s) you used.
Bonus: Your three expressions must not be the same as each other and must include at least one example of each of $\times$, $\div$, and $+ / -$.
How did this worksheet feel?
What I'll revisit before next class:
1.1 — $(5 \times 10^{-2})(3 \times 10^6)$
Coefficients: $5 \times 3 = 15$. Indices: $10^{-2 + 6} = 10^4$. Current: $15 \times 10^4$ — coefficient out of $[1, 10)$. Re-standardise: $\mathbf{1.5 \times 10^5}$.
1.2 — $\dfrac{8 \times 10^6}{2 \times 10^2} - 1 \times 10^3$
Quotient: $\dfrac{8}{2} \times 10^{6 - 2} = 4 \times 10^4$.
Subtract: $4 \times 10^4 - 1 \times 10^3$. Match powers: $1 \times 10^3 = 0.1 \times 10^4$. Then $(4 - 0.1) \times 10^4 = \mathbf{3.9 \times 10^4}$.
1.3 — $6 \times 10^{-2} - 5 \times 10^{-3}$
Match powers to larger index ($-2$): $5 \times 10^{-3} = 0.5 \times 10^{-2}$. Subtract coefficients: $6 - 0.5 = 5.5$. Answer: $\mathbf{5.5 \times 10^{-2}}$.
1.4 — Verify $(2.5 \times 10^4) \times (4 \times 10^{-7}) = 1 \times 10^{-2}$
Coefficients: $2.5 \times 4 = 10$. Indices: $10^{4 + (-7)} = 10^{-3}$. Current: $10 \times 10^{-3}$. Re-standardise: $10 \times 10^{-3} = 1.0 \times 10^{-2}$. Verified — answer is $\mathbf{1 \times 10^{-2}}$ (also written $0.01$).
1.5 — Solve $(4 \times 10^n)(2 \times 10^3) = 8 \times 10^7$
Coefficients: $4 \times 2 = 8$ ✓ (matches RHS). Indices: $10^{n + 3} = 10^7$, so $n + 3 = 7$, giving $\mathbf{n = 4}$.
Check: $(4 \times 10^4)(2 \times 10^3) = 8 \times 10^{4 + 3} = 8 \times 10^7$ ✓.
1.6 — $\dfrac{(6 \times 10^5)(5 \times 10^{-2})}{3 \times 10^2}$
Step 1 — numerator with $\times$: $(6 \times 5) \times 10^{5 + (-2)} = 30 \times 10^3 = 3 \times 10^4$ (re-standardised).
Step 2 — divide by denominator: $\dfrac{3 \times 10^4}{3 \times 10^2} = 1 \times 10^{4 - 2} = \mathbf{1 \times 10^2}$.
2 — Find the mistake
(a) The mistake is on Line 3.
(b) The student added the powers of 10 together ($10^5 + 10^5 = 10^{10}$), but once the powers match, the $10^5$ is just a common factor — it doesn't get added or multiplied. Only the coefficients combine: $(4 + 0.3) \times 10^5$.
(c) Corrected working:
$4 \times 10^5 + 3 \times 10^4$
$= 4 \times 10^5 + 0.3 \times 10^5$ (common power $10^5$)
$= (4 + 0.3) \times 10^5$
$= \mathbf{4.3 \times 10^5}$.
This is exactly the trap flagged in the lesson's "Spot the Trap" and "Common Pitfalls" cards: adding indices when adding terms.
3 — Open-ended challenge (sample solution)
We need to build $6 \times 10^7$ using each of $\times$, $\div$ and $+ / -$ at least once across the three expressions.
Expression 1 (using $\times$): $(2 \times 10^3)(3 \times 10^4)$.
Working: $(2 \times 3) \times 10^{3 + 4} = 6 \times 10^7$ ✓.
Rules used: multiply coefficients, add indices (product rule on powers of 10).
Expression 2 (using $\div$): $\dfrac{1.2 \times 10^{10}}{2 \times 10^2}$.
Working: $\dfrac{1.2}{2} \times 10^{10 - 2} = 0.6 \times 10^8 = 6 \times 10^7$ ✓.
Rules used: divide coefficients, subtract indices (quotient rule), then re-standardise.
Expression 3 (using $+$): $5 \times 10^7 + 1 \times 10^7$.
Working: powers already match, $(5 + 1) \times 10^7 = 6 \times 10^7$ ✓.
Rules used: matching powers first, then add coefficients.
Other valid examples: $\dfrac{3 \times 10^9}{5 \times 10^1}$, $(4 \times 10^7) + (2 \times 10^7)$, $(6 \times 10^4)(1 \times 10^3)$.
Marking: 1 mark per valid expression with shown working (max 3). 1 mark for using all three operation types across the set.