Mathematics • Year 9 • Unit 1 • Lesson 18
Applications of Scientific Notation
Build fluency with multiplying and dividing in scientific notation, comparing cosmic and atomic numbers, rounding to significant figures, and using the EE/EXP key on a calculator. Every answer ends in standard form $a \times 10^n$ with $1 \le a < 10$.
1. I do — fully worked example
Read every line. Each step has a short reason on the right so you can see why, not just what.
Problem. Calculate $(4 \times 10^5) \times (3 \times 10^6)$ and write the answer in scientific notation.
Step 1 — Spot the rule.
Multiplication of two $a \times 10^n$ numbers $\to$ multiply coefficients, add indices.
Reason: $(a \times 10^m)(b \times 10^n) = (a b) \times 10^{m+n}$. Ordinary arithmetic on coefficients, product rule on powers of 10.
Step 2 — Multiply the coefficients.
$4 \times 3 = 12$
Reason: coefficients are just numbers — multiply them normally.
Step 3 — Add the indices.
$10^5 \times 10^6 = 10^{5 + 6} = 10^{11}$
Reason: product rule on the powers of 10 — add the indices.
Step 4 — Combine.
$12 \times 10^{11}$
Reason: $12$ is outside $[1, 10)$ — value is right, form is wrong.
Step 5 — Re-normalise into standard form.
$12 \times 10^{11} = 1.2 \times 10^{12}$ (decimal shifts one left $\to$ index goes up by 1)
Reason: keep $1 \le a < 10$. Slide the decimal one place; nudge the index by $\pm 1$ in the opposite direction.
Answer: $\mathbf{1.2 \times 10^{12}}$.
2. We do — fill in the missing steps
Same structure as Section 1, with the working faded. Fill in each blank. 4 marks
Problem. Calculate $\dfrac{9 \times 10^8}{3 \times 10^{-2}}$.
Step 1 — Spot the rule: division of two $a \times 10^n$ numbers $\to$ divide the coefficients and __________________ the indices.
Step 2 — Divide the coefficients:
$\dfrac{9}{3} = \_\_\_\_$
Step 3 — Subtract the indices (watch the negative!):
$10^{8 - (\_\_\_\_)} = 10^{8 + \_\_\_\_} = 10^{\_\_\_\_}$
Step 4 — Combine:
$\dfrac{9 \times 10^8}{3 \times 10^{-2}} = \_\_\_\_\_\_\_\_\_$
Step 5 — Check the form: Is the coefficient in $[1, 10)$? __________ (yes / no)
3. You do — independent practice
Show your working under each problem. The first four are foundation. The middle two are standard. The last two are extension.
Foundation — single rule
3.1 Calculate $(2 \times 10^7) \times (4 \times 10^3)$. 1 mark
3.2 Calculate $\dfrac{8 \times 10^{12}}{2 \times 10^{-4}}$. 1 mark
3.3 Round $4.6738 \times 10^{-5}$ to 3 significant figures. 1 mark
3.4 Which is larger: $2 \times 10^{10}$ or $9 \times 10^9$? Justify in one line. 1 mark
Standard — re-normalise required
3.5 Calculate $(6 \times 10^4) \times (5 \times 10^7)$, giving the answer in standard form. 2 marks
3.6 Calculate $\dfrac{1.5 \times 10^{11}}{5 \times 10^{-11}}$, giving the answer in standard form. 2 marks
Extension — push your thinking
3.7 The world's population is about $8.0 \times 10^9$. The average arm-span is $1.5$ m. If everyone held hands in one line, what total length is that, to 2 significant figures? 2 marks
3.8 A student types $3 \times 10$ EXP $8$ on their calculator and gets $3 \times 10^9$. Explain in one sentence what they did wrong and what they should have typed. 2 marks
How did this worksheet feel?
What I'll revisit before next class:
Section 2 — We do (faded $\dfrac{9 \times 10^8}{3 \times 10^{-2}}$)
Step 1: division $\to$ subtract indices.
Step 2: $\dfrac{9}{3} = \mathbf{3}$.
Step 3: $10^{8 - (\mathbf{-2})} = 10^{8 + \mathbf{2}} = 10^{\mathbf{10}}$.
Step 4: $\mathbf{3 \times 10^{10}}$.
Step 5: $3$ is in $[1, 10)$ — yes, in standard form.
3.1 — $(2 \times 10^7) \times (4 \times 10^3)$
$2 \times 4 = 8$; $10^{7 + 3} = 10^{10}$. Answer: $\mathbf{8 \times 10^{10}}$.
3.2 — $\dfrac{8 \times 10^{12}}{2 \times 10^{-4}}$
$8 / 2 = 4$; $10^{12 - (-4)} = 10^{16}$. Answer: $\mathbf{4 \times 10^{16}}$.
3.3 — Round $4.6738 \times 10^{-5}$ to 3 s.f.
Keep first three digits: $4$, $6$, $7$. Next digit is $3 < 5$, so round down. Answer: $\mathbf{4.67 \times 10^{-5}}$.
3.4 — Compare $2 \times 10^{10}$ and $9 \times 10^9$
$\mathbf{2 \times 10^{10}}$ is larger. The index $10$ is greater than $9$, and bigger index wins for positive coefficients — the coefficient only matters when the indices tie.
3.5 — $(6 \times 10^4) \times (5 \times 10^7)$
$6 \times 5 = 30$; $10^{4 + 7} = 10^{11}$. Current: $30 \times 10^{11}$. Re-normalise: $\mathbf{3 \times 10^{12}}$ (decimal one left, index up by 1).
3.6 — $\dfrac{1.5 \times 10^{11}}{5 \times 10^{-11}}$
$\dfrac{1.5}{5} = 0.3$; $10^{11 - (-11)} = 10^{22}$. Current: $0.3 \times 10^{22}$. Re-normalise: $\mathbf{3 \times 10^{21}}$ (decimal one right, index down by 1).
3.7 — World population arm-span
Length $= (8.0 \times 10^9) \times 1.5 = (8.0 \times 1.5) \times 10^9 = 12 \times 10^9 = \mathbf{1.2 \times 10^{10}}$ m (2 s.f.).
That's about 12 billion metres — over 30 times the distance to the Moon.
3.8 — Calculator mistake
The EE / EXP key is the "$\times 10^{?}$" — so typing $\times 10$ before pressing it adds an extra factor of $10$, giving $3 \times 10 \times 10^8 = 3 \times 10^9$. The correct key sequence is just "$3$ EXP $8$" to enter $\mathbf{3 \times 10^8}$.