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Amy Ferguson Moncure

IB · Unit 1 · 1B Scientific Notation

Standard Form Applications

Use standard form on measurements: how many times as big, people per area, and volumes.

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Key Ideas IB wording Calculator Easy Medium Hard

\(x^2 \cdot x^3 = x^5\)

\((x^2)^3 = x^6\)

3−2 = 19

What to Remember

Same operations

Word problems still use multiply, divide, add, or subtract in standard form. Convert units if the problem mixes them.

How many times as big

Divide the larger quantity by the smaller one. Rewrite the quotient in standard form.

People per unit area

Population ÷ area. Match the units first (m² with m², km² with km²).

Sphere volume

V = 43πr³ and r is half the diameter. Cube the standard-form radius by cubing the coefficient and multiplying the exponent by 3.

IB wording = scientific notation

When a question says write the answer in the form a × 10ᵏ where 1 ≤ a < 10 and k ⊂ ℤ, that is the IB way of saying write the answer in scientific notation.

k ⊂ ℤ means k is an integer (k is in the set of integers).

TI-84 Plus CE: turn on SCI
  1. Press MODE.
  2. On the first line (NORMAL SCI ENG), arrow right to SCI.
  3. Press ENTER so SCI is highlighted.
  4. Press 2nd then MODE (QUIT) to return to the home screen.

Answers will look like 3.2E4. That means 3.2 × 10⁴. Write the × 10 form on paper.

Level 1

Easy

Read the measurement and write it as an ordinary number or in standard form.

Example
A lake has area 4.6 × 10⁷ m². Write this as an ordinary number.

Move the decimal 7 places right.

46 000 000 m²

Answer: 46 000 000 m²

1
A virus is 2.5 × 10⁻⁷ m long. Write this as a decimal.
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Move the decimal 7 places left.

0.00000025 m

Answer: 0.00000025 m

2
IB Standard form (scientific notation). 0.00000081 seconds.
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Move the decimal 7 places right.

8.1 × 10⁻⁷ s

Answer: 8.1 × 10⁻⁷ s

3
A city has 3.8 × 10⁶ people. Write this as an ordinary number.
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3 800 000 people

Answer: 3 800 000 people

4
IB Standard form (scientific notation). 92 000 000 km.
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9.2 × 10⁷ km

Answer: 9.2 × 10⁷ km

5
A battery stores 1.2 × 10⁻³ kWh. Write this as a decimal.
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0.0012 kWh

Answer: 0.0012 kWh

Level 2

Medium

One multiply or divide, then write the answer in standard form.

Example
A pebble has mass 8 × 10⁻¹ g. A dust particle has mass 3.2 × 10⁻⁶ g. How many times heavier is the pebble?

8 × 10⁻¹3.2 × 10⁻⁶ = 2.5 × 10⁵

The pebble is 2.5 × 10⁵ times heavier.

Answer: 2.5 × 10⁵ times heavier

1
Light travels at 3.00 × 10⁸ m/s. How far does it travel in 2 × 10³ s?
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3.00 · 2 = 6.00. Exponents 8 + 3 = 11.

6.00 × 10¹¹ m

Answer: 6.00 × 10¹¹ m

2
Two parks have areas 1.2 × 10⁶ m² and 4 × 10⁵ m². Find the total area.
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4 × 10⁵ = 0.4 × 10⁶.

1.2 + 0.4 = 1.6.

1.6 × 10⁶ m²

Answer: 1.6 × 10⁶ m²

3
A town has 7.4 × 10⁶ people and area 2 × 10³ km². Find the number of people per km².
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7.4 ÷ 2 = 3.7. Exponents 6 − 3 = 3.

3.7 × 10³ people/km²

Answer: 3.7 × 10³ people/km²

4
A beach has 4.8 × 10¹² grains of sand. One pile holds 6 × 10⁸ grains. How many piles is that?
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4.8 ÷ 6 = 0.8. Exponents 12 − 8 = 4.

0.8 × 10⁴ = 8 × 10³

Answer: 8 × 10³ piles

5
A machine makes 4.5 × 10³ parts per hour for 8 × 10¹ hours. How many parts is that?
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4.5 · 8 = 36. Exponents 3 + 1 = 4.

36 × 10⁴ = 3.6 × 10⁵

Answer: 3.6 × 10⁵ parts

Level 3

Hard

More than one step. Keep every answer in standard form with 1 ≤ a < 10.

Example
A sample has mass 15 g. 30 g of this element contains 6.0 × 10²³ atoms. How many atoms are in the 15 g sample?

15 g is half of 30 g, so take half the atoms.

12 × 6.0 × 10²³ = 3.0 × 10²³

Answer: 3.0 × 10²³ atoms

1
A spherical droplet has diameter 6 × 10⁻⁶ m. Using V = 43πr³, find the volume in terms of π.
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r = 3 × 10⁻⁶.

r³ = 27 × 10⁻¹⁸ = 2.7 × 10⁻¹⁷

V = 43π · 2.7 × 10⁻¹⁷ = 3.6π × 10⁻¹⁷

3.6π × 10⁻¹⁷ m³

Answer: 3.6π × 10⁻¹⁷ m³

2
Country A has area 2.5 × 10¹² m² and population 5.0 × 10⁷. Country B has area 8.0 × 10¹¹ m² and population 2.0 × 10⁸. Which country has more people per square metre?
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A: 5.0 × 10⁷2.5 × 10¹² = 2.0 × 10⁻⁵ people/m².

B: 2.0 × 10⁸8.0 × 10¹¹ = 2.5 × 10⁻⁴ people/m².

Country B has more people per square metre.

Answer: Country B (2.5 × 10⁻⁴ vs 2.0 × 10⁻⁵ people/m²)

3
A floor uses 2.0 × 10⁴ tiles. Each tile has area 5.0 × 10⁻² m². Find the total area.
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2.0 · 5.0 = 10. Exponents 4 + (−2) = 2.

10 × 10² = 1.0 × 10³ m²

Answer: 1.0 × 10³ m²

4
The Earth–Sun distance is about 1.5 × 10¹¹ m. Light travels at 3.0 × 10⁸ m/s. How many seconds does the light take?
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1.5 × 10¹¹3.0 × 10⁸ = 0.5 × 10³ = 5.0 × 10²

5.0 × 10² s

Answer: 5.0 × 10² s

5
A cube has side 4 × 10⁻³ m. Find its volume in standard form.
Show Answer

V = s³ = 64 × 10⁻⁹

64 × 10⁻⁹ = 6.4 × 10⁻⁸ m³

Answer: 6.4 × 10⁻⁸ m³

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