Operations with Scientific Notation (A) Worksheet | Fun and ... - Free Printable

Here is the complete solution to the worksheet “Operations with Scientific Notation (A)”, solved step-by-step and explained clearly.

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## ✔ Section A: Multiplication and Division

Rules to remember:
- Multiply coefficients → add exponents.
- Divide coefficients → subtract exponents.
- Adjust final answer to proper scientific notation: coefficient between 1 and 10.

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1) `(3 × 10⁶) × (4 × 10²)`

→ Multiply coefficients: `3 × 4 = 12`
→ Add exponents: `10⁶⁺² = 10⁸`
→ `12 × 10⁸` → adjust: `1.2 × 10⁹`

✔ Answer: `1.2 × 10⁹`

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2) `(7 × 10⁶) × (3 × 10¹⁸)`

→ `7 × 3 = 21`
→ `10⁶⁺¹⁸ = 10²⁴`
→ `21 × 10²⁴ → 2.1 × 10²⁵`

✔ Answer: `2.1 × 10²⁵`

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3) `(6 × 10⁻¹¹) × (4 × 10²⁸)`

→ `6 × 4 = 24`
→ `10⁻¹¹⁺²⁸ = 10¹⁷`
→ `24 × 10¹⁷ → 2.4 × 10¹⁸`

✔ Answer: `2.4 × 10¹⁸`

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4) `(1.2 × 10⁻³) × (7 × 10⁻¹⁰)`

→ `1.2 × 7 = 8.4`
→ `10⁻³⁻¹⁰ = 10⁻¹³`
→ Already in scientific notation.

✔ Answer: `8.4 × 10⁻¹³`

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5) `(2.5 × 10⁹) × (8 × 10²⁰)`

→ `2.5 × 8 = 20`
→ `10⁹⁺²⁰ = 10²⁹`
→ `20 × 10²⁹ → 2.0 × 10³⁰`

✔ Answer: `2.0 × 10³⁰`

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6) `(1.1 × 10⁻⁴) × (9 × 10¹³)`

→ `1.1 × 9 = 9.9`
→ `10⁻⁴⁺¹³ = 10⁹`
→ Already in scientific notation.

✔ Answer: `9.9 × 10⁹`

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7) `(2.4 × 10⁻⁸) ÷ (4 × 10⁵⁵)`

→ Divide coefficients: `2.4 ÷ 4 = 0.6`
→ Subtract exponents: `10⁻⁸⁻⁵⁵ = 10⁻⁶³`
→ `0.6 × 10⁻⁶³ → 6.0 × 10⁻⁶⁴`

✔ Answer: `6.0 × 10⁻⁶⁴`

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8) `(8.6 × 10³⁹) ÷ (2 × 10⁻²⁰)`

→ `8.6 ÷ 2 = 4.3`
→ `10³⁹⁻⁽⁻²⁰⁾ = 10³⁹⁺²⁰ = 10⁵⁹`
→ Already in scientific notation.

✔ Answer: `4.3 × 10⁵⁹`

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## ✔ Section B: Powers, Addition, Subtraction

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1) `(5 × 10³)²`

→ Square coefficient: `5² = 25`
→ Double exponent: `(10³)² = 10⁶`
→ `25 × 10⁶ → 2.5 × 10⁷`

✔ Answer: `2.5 × 10⁷`

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2) `(7 × 10⁴)²`

→ `7² = 49`
→ `(10⁴)² = 10⁸`
→ `49 × 10⁸ → 4.9 × 10⁹`

✔ Answer: `4.9 × 10⁹`

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3) `(3 × 10⁻³)³`

→ `3³ = 27`
→ `(10⁻³)³ = 10⁻⁹`
→ `27 × 10⁻⁹ → 2.7 × 10⁻⁸`

✔ Answer: `2.7 × 10⁻⁸`

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4) `(2 × 10⁻²)⁻²`

→ Coefficient: `2⁻² = 1/4 = 0.25`
→ Exponent: `(10⁻²)⁻² = 10⁴`
→ `0.25 × 10⁴ → 2.5 × 10³`

✔ Answer: `2.5 × 10³`

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5) `(9 × 10³) + (3 × 10⁵)`

→ Make exponents same: convert `9 × 10³ = 0.09 × 10⁵`
→ Add: `0.09 × 10⁵ + 3 × 10⁵ = 3.09 × 10⁵`

✔ Answer: `3.09 × 10⁵`

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6) `(2.1 × 10⁻⁴) + (3.04 × 10⁻⁵)`

→ Convert to same exponent: `3.04 × 10⁻⁵ = 0.304 × 10⁻⁴`
→ Add: `2.1 × 10⁻⁴ + 0.304 × 10⁻⁴ = 2.404 × 10⁻⁴`

✔ Answer: `2.404 × 10⁻⁴`

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7) `(7.05 × 10⁷) - (4.807 × 10⁵)`

→ Convert: `4.807 × 10⁵ = 0.04807 × 10⁷`
→ Subtract: `7.05 - 0.04807 = 7.00193`
→ Answer: `7.00193 × 10⁷`

✔ Answer: `7.00193 × 10⁷`

*(Note: Depending on significant figures, you might round, but problem doesn’t specify — so keep as is.)*

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8) `(4.06 × 10⁻⁶) - (9.89 × 10⁻⁷)`

→ Convert: `9.89 × 10⁻⁷ = 0.989 × 10⁻⁶`
→ Subtract: `4.06 - 0.989 = 3.071`
→ Answer: `3.071 × 10⁻⁶`

✔ Answer: `3.071 × 10⁻⁶`

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## ✔ Section C: Word Problems

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1) California GDP in 2014 = `$2.3 × 10¹²`

#### a) 11% from manufacturing
→ `11% of 2.3 × 10¹² = 0.11 × 2.3 × 10¹²`
→ `0.11 × 2.3 = 0.253`
→ So, `0.253 × 10¹² = 2.53 × 10¹¹`

✔ Answer: `$2.53 × 10¹¹`

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#### b) Population increase from 1994 to 2014:
- 1994: `3.15 × 10⁷`
- 2014: `3.88 × 10⁷`

→ Increase = `3.88 × 10⁷ - 3.15 × 10⁷ = 0.73 × 10⁷ = 7.3 × 10⁶`
→ % increase = `(Increase / Original) × 100`
= `(7.3 × 10⁶) / (3.15 × 10⁷) × 100`
= `(7.3 / 31.5) × 100 ≈ 0.2317 × 100 ≈ 23.17%`

✔ Answer: `23.17%`

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2) Radius of Jupiter = `7.149 × 10⁴ km`

#### a) Circumference = `2πr`
→ Use π ≈ 3.1416
→ `2 × 3.1416 × 7.149 × 10⁴`
→ First: `2 × 3.1416 = 6.2832`
→ `6.2832 × 7.149 ≈ ?`

Let’s compute:

`6.2832 × 7 = 43.9824`
`6.2832 × 0.149 ≈ 6.2832 × 0.15 = 0.94248` minus `6.2832 × 0.001 = 0.0062832` → approx `0.9362`
Total ≈ `43.9824 + 0.9362 = 44.9186`

→ So, circumference ≈ `44.9186 × 10⁴ km = 4.49186 × 10⁵ km`

To 3 significant figures: `4.49 × 10⁵ km`

✔ Answer: `4.49 × 10⁵ km`

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#### b) Time to travel around Jupiter at 120 mph

First, convert circumference to miles:

Given: `1 mile = 1.6 km`
→ Circumference in miles = `(4.49186 × 10⁵) ÷ 1.6`
→ `4.49186 ÷ 1.6 ≈ 2.8074`
→ So, `2.8074 × 10⁵ miles`

Speed = 120 miles per hour

Time (hours) = Distance ÷ Speed = `(2.8074 × 10⁵) ÷ 120`
→ `2.8074 × 10⁵ ÷ 120 = (2.8074 ÷ 120) × 10⁵`
→ `2.8074 ÷ 120 ≈ 0.023395`
→ So, `0.023395 × 10⁵ = 2.3395 × 10³ hours`

Convert to days: divide by 24
→ `2.3395 × 10³ ÷ 24 ≈ 97.479 days`

✔ Answer: Approximately `97.5 days` (rounded to 3 sig figs)

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## 📌 Final Answers Summary:

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Section A

1) `1.2 × 10⁹`
2) `2.1 × 10²⁵`
3) `2.4 × 10¹⁸`
4) `8.4 × 10⁻¹³`
5) `2.0 × 10³⁰`
6) `9.9 × 10⁹`
7) `6.0 × 10⁻⁶⁴`
8) `4.3 × 10⁵⁹`

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Section B

1) `2.5 × 10⁷`
2) `4.9 × 10⁹`
3) `2.7 × 10⁻⁸`
4) `2.5 × 10³`
5) `3.09 × 10⁵`
6) `2.404 × 10⁻⁴`
7) `7.00193 × 10⁷`
8) `3.071 × 10⁻⁶`

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Section C

1a) `$2.53 × 10¹¹`
1b) `23.17%`
2a) `4.49 × 10⁵ km`
2b) `97.5 days`

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✔ All problems solved without calculator, answers in scientific notation where required.

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