Lesson 18 ~25 min Unit 1 · Index Laws +85 XP

Applications of Scientific Notation

Compare cosmic and atomic numbers, multiply and divide in scientific notation, round to significant figures, and use the EE/EXP key on your calculator.

Today's hook: The Sun is about $1.5 \times 10^{11}$ m away. A hydrogen atom is about $5 \times 10^{-11}$ m wide. How many atoms fit end-to-end along that distance — without a single zero in your working?

0/5QUESTS

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Think First

warm-up

The mass of the Earth is about $6 \times 10^{24}$ kg. The mass of a person is about $6 \times 10^{1}$ kg. About how many people would weigh the same as the Earth? Estimate the index first — then check the coefficient.

Record in your workbook.

The Big Idea

+5 XP

Scientific notation lets us multiply and divide colossal or tiny numbers using just the index laws — coefficients use ordinary arithmetic, and the powers of 10 use the product and quotient rules.

$(a \times 10^m) \times (b \times 10^n) = (a \times b) \times 10^{m+n}$ — multiply coefficients, add indices. $\dfrac{a \times 10^m}{b \times 10^n} = \dfrac{a}{b} \times 10^{m-n}$ — divide coefficients, subtract indices. Adjust at the end so the coefficient sits between $1$ and $10$.

$(3 \times 10^8) \times (2 \times 10^5) = 6 \times 10^{13}$

Multiply: add powers

$10^m \times 10^n = 10^{m+n}$.

Divide: subtract powers

$10^m / 10^n = 10^{m-n}$.

Tidy the coefficient

Make $1 \le a < 10$.

What You'll Master

objectives

Know

Standard form $a \times 10^n$ where $1 \le a < 10$
Multiply / divide rules in scientific notation
Meaning of significant figures (sig fig)

Understand

Why scientific notation makes huge / tiny numbers comparable
How to re-adjust a coefficient that drifts out of $[1, 10)$
Why 3 sig fig is the usual answer precision in science

Can Do

Compute $(3 \times 10^8) \times (2 \times 10^5)$
Compare $1.5 \times 10^{11}$ to $5 \times 10^{-11}$
Use the EE / EXP key on a scientific calculator

Words You Need

vocabulary

Scientific notation$a \times 10^n$, with $1 \le a < 10$ and integer $n$.

Coefficient (mantissa)The $a$ part — carries the digits.

Order of magnitudeThe power of 10. A jump of $1$ in $n$ means $10\times$ bigger.

Significant figuresThe non-leading-zero digits that carry information. $0.00420$ has $3$ sig fig.

EE / EXP keyCalculator key for $\times 10^?$. Type the index after pressing it.

Re-normaliseMove the decimal so the coefficient is back in $[1, 10)$ and update the power.

Spot the Trap

heads-up

Wrong: “$(4 \times 10^5) \times (3 \times 10^6) = 12 \times 10^{11}$” left as-is — the coefficient $12$ is not in $[1, 10)$.

Right: $12 \times 10^{11} = 1.2 \times 10^{12}$. Tidy the coefficient.

Wrong: Typing “$3 \times 10$ EXP $8$” on the calculator — you've entered $3 \times 10 \times 10^8 = 3 \times 10^9$.

Right: Type “$3$ EXP $8$”. The EXP key is the $\times 10^?$ part already.

Comparing very large & very small

+5 XP

Compare the powers of 10 first. Whichever has the larger index is bigger (for positive coefficients). Only if the indices tie do you compare the coefficients.

Which is bigger: $7 \times 10^9$ or $2 \times 10^{10}$? The indices are $9$ and $10$, so $2 \times 10^{10}$ wins — even though its coefficient is smaller. How many times bigger? $\dfrac{2 \times 10^{10}}{7 \times 10^9} = \dfrac{2}{7} \times 10^{1} \approx 2.86$ times bigger.

$2 \times 10^{10} \approx 2.86 \times (7 \times 10^9)$

Significant figures — an introduction

+5 XP

Sig figs count the digits that carry information. Leading zeros never count; trailing zeros after a decimal point do count. Most science answers are quoted to 3 sig fig.

$0.00420$ has $3$ sig fig (the digits $4$, $2$, $0$). $5{,}730{,}000$ written as $5.73 \times 10^6$ shows $3$ sig fig clearly. Round $2.4763 \times 10^8$ to $3$ sig fig: look at the 4th digit ($6 \ge 5$), so round up — $2.48 \times 10^8$.

$2.4763 \times 10^8 \approx 2.48 \times 10^8$ (3 s.f.)

Watch Me Solve It · 3 examples

Watch Me Solve It · Multiply with re-normalise

+15 XP per step

PROBLEM

Calculate $(4 \times 10^5) \times (3 \times 10^6)$ and write the answer in scientific notation.

1
Multiply the coefficients
$4 \times 3 = 12$
2
Add the indices (product rule on powers of 10)
$10^5 \times 10^6 = 10^{11}$
3
Re-normalise so $1 \le a < 10$
$12 \times 10^{11} = 1.2 \times 10^{12}$
Decimal shifts one left $\to$ index goes up by 1.

Answer$1.2 \times 10^{12}$

Watch Me Solve It · How many atoms fit?

+15 XP per step

PROBLEM

How many hydrogen atoms (each $5 \times 10^{-11}$ m wide) would line up across the Earth–Sun distance ($1.5 \times 10^{11}$ m)?

1
Set up as a quotient
$N = \dfrac{1.5 \times 10^{11}}{5 \times 10^{-11}}$
2
Divide coefficients, subtract indices
$\dfrac{1.5}{5} = 0.3$; $10^{11 - (-11)} = 10^{22}$
3
Re-normalise
$0.3 \times 10^{22} = 3 \times 10^{21}$
Coefficient shifts right by one $\to$ index drops by 1.

Answer$3 \times 10^{21}$ atoms

Watch Me Solve It · Round to 3 sig fig

+15 XP per step

PROBLEM

A bacterium has mass $9.5 \times 10^{-13}$ g. How many fit into $1$ gram, to 3 sig fig?

1
Write the quotient
$N = \dfrac{1}{9.5 \times 10^{-13}}$
2
Divide $1$ by the coefficient
$\dfrac{1}{9.5} \approx 0.10526$
3
Flip the index sign (dividing by $10^{-13}$ = $\times 10^{13}$) and re-normalise to 3 s.f.
$0.10526 \times 10^{13} = 1.0526 \times 10^{12} \approx 1.05 \times 10^{12}$

Answer$\approx 1.05 \times 10^{12}$ bacteria

Common Pitfalls

heads-up

Leaving a coefficient outside $[1, 10)$

$12 \times 10^{11}$ and $0.3 \times 10^{22}$ are not standard form.

Fix: Slide the decimal one place; nudge the index by $\pm 1$ in the opposite direction.

Typing $\times 10$ before the EE key

$3 \times 10$ EE $8$ gives $3 \times 10^9$, not $3 \times 10^8$.

Fix: Just type the coefficient, press EE / EXP, then the index.

Forgetting signs on indices

$10^{11} \div 10^{-11}$ is $10^{22}$, not $10^0$.

Fix: Quotient rule subtracts. Subtracting a negative adds.

Copy Into Your Books

Multiply

$(a \times 10^m)(b \times 10^n) = ab \times 10^{m+n}$
Re-normalise after

Divide

$\dfrac{a \times 10^m}{b \times 10^n} = \dfrac{a}{b} \times 10^{m-n}$
Subtract indices carefully with signs

Compare

Bigger index = bigger number
Same index $\to$ compare coefficients

Significant figures

3 s.f. is standard in science
Round the digit after the last kept digit

How are you completing this lesson?

Brain Trainer · 4 problems

Brain Trainer · Scientific notation drills

4 problems

Mix multiplication, division and a real-world question. Round answers to 3 sig fig when asked.

1 Calculate $(2 \times 10^7) \times (4 \times 10^3)$.
$2 \times 4 = 8$; $10^{7+3} = 10^{10}$.$8 \times 10^{10}$
2 Calculate $\dfrac{8 \times 10^{12}}{2 \times 10^{-4}}$.
$8 / 2 = 4$; $10^{12 - (-4)} = 10^{16}$.$4 \times 10^{16}$
3 The world's population is about $8.0 \times 10^9$. If everyone held $1.5$ m of arm-span, what total length is that, to 2 s.f.?
$8.0 \times 1.5 = 12.0$; $\times 10^9$ m $= 1.2 \times 10^{10}$ m.$1.2 \times 10^{10}$ m
4 Round $4.6738 \times 10^{-5}$ to 3 sig fig.
Keep first 3 digits: $4$, $6$, $7$. Next digit $3 < 5$, so round down.$4.67 \times 10^{-5}$

Complete in your workbook.

Quick Check · 5 questions

Calculate $(5 \times 10^6) \times (4 \times 10^3)$.

+10 XP

Calculate $\dfrac{9 \times 10^8}{3 \times 10^{-2}}$.

+10 XP

Which is largest?

+10 XP

Round $3.05471 \times 10^7$ to 3 significant figures.

+10 XP

Light travels at $3.0 \times 10^8$ m/s. How long (in seconds) to reach Earth from the Sun ($1.5 \times 10^{11}$ m away)?

+10 XP

Show Your Working · 3 questions

Show Your Working

9 marks total

ApplyMedium3 MARKS

Q6. Calculate, leaving the answer in scientific notation: (a) $(6 \times 10^4) \times (5 \times 10^7)$, (b) $\dfrac{4.8 \times 10^{-3}}{1.6 \times 10^{2}}$, (c) $(2 \times 10^5)^3$.

Answer in your workbook.

ApplyMedium3 MARKS

Q7. The mass of an electron is $9.11 \times 10^{-31}$ kg. The mass of a proton is $1.67 \times 10^{-27}$ kg. (a) Which is heavier? (b) How many times heavier, to 3 sig fig?

Answer in your workbook.

ReasonHard3 MARKS

Q8. Australia's population is about $2.6 \times 10^7$. The federal budget is about $\$6.5 \times 10^{11}$. Calculate the spending per person, to 3 sig fig. Show your calculator key-sequence using EE / EXP.

Answer in your workbook.

Comprehensive Answers

Quick Check

1. B — $2 \times 10^{10}$.

2. C — $3 \times 10^{10}$.

3. A — $2 \times 10^{10}$.

4. D — $3.05 \times 10^7$.

5. B — $5 \times 10^{2}$ s ($\approx 500$ s, about 8.3 minutes).

Show Your Working Model Answers

Q6 (3 marks): (a) $6 \times 5 = 30$, $10^{4+7} = 10^{11}$, re-normalise: $3 \times 10^{12}$ [1]; (b) $4.8/1.6 = 3$, $10^{-3-2} = 10^{-5}$, so $3 \times 10^{-5}$ [1]; (c) $2^3 = 8$, $10^{5 \times 3} = 10^{15}$, so $8 \times 10^{15}$ [1].

Q7 (3 marks): (a) Indices: $-27 > -31$, so the proton is heavier [1]. (b) Ratio = $\dfrac{1.67}{9.11} \times 10^{-27 - (-31)} = 0.1833 \times 10^{4} = 1.833 \times 10^{3}$ [1], i.e. about $1{,}830$ times heavier ($\approx 1.83 \times 10^3$ to 3 s.f.) [1].

Q8 (3 marks): Keys: $6.5$ EXP $11$ $\div$ $2.6$ EXP $7$ $=$ [1]. Coefficient: $6.5 / 2.6 = 2.500$ [1]. Index: $10^{11-7} = 10^4$. Answer: $2.50 \times 10^4 = \$25{,}000$ per person (3 s.f.) [1].

Stretch Challenge · +25 XP, +10 coins

Atomic Headcount

A single grain of table salt contains roughly $6.0 \times 10^{18}$ formula units of NaCl. If you eat $5.0$ g of salt at dinner (one grain $\approx 5.85 \times 10^{-5}$ g), about how many formula units have you swallowed? Give your answer in scientific notation to 2 sig fig, then state the order of magnitude.

Reveal solution

Grains: $\dfrac{5.0}{5.85 \times 10^{-5}} \approx 8.547 \times 10^{4}$. Formula units: $8.547 \times 10^{4} \times 6.0 \times 10^{18} \approx 51.28 \times 10^{22} = 5.1 \times 10^{23}$. Order of magnitude $\approx 10^{23}$.

Quick Review

Multiply

$ab \times 10^{m+n}$

Divide

$\dfrac{a}{b} \times 10^{m-n}$

Compare

Index first, then coefficient

Re-normalise

Keep $1 \le a < 10$

3 sig fig

Standard for science

EE / EXP

Replaces $\times 10^?$

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