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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