Operations with Scientific Notation Worksheets - Free Printable

Sure! Let’s solve each of these 10 problems involving operations with scientific notation, and express each answer in scientific notation.

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Key Rules to Remember:

- Multiplication: Multiply coefficients, add exponents.
`(a × 10^m) × (b × 10^n) = (a×b) × 10^(m+n)`
- Division: Divide coefficients, subtract exponents.
`(a × 10^m) ÷ (b × 10^n) = (a/b) × 10^(m−n)`
- Addition/Subtraction: Must have the same exponent. Adjust one number so exponents match, then add/subtract coefficients.
- Scientific Notation Format: `a × 10^n`, where `1 ≤ |a| < 10` and `n` is an integer.

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## Problem 1: `(7 × 10⁴)(2 × 10⁶)`

Multiply coefficients: `7 × 2 = 14`
Add exponents: `4 + 6 = 10`
→ `14 × 10¹⁰`

But 14 is not between 1 and 10 → convert:
`14 × 10¹⁰ = 1.4 × 10¹¹`

✔ Answer: `1.4 × 10¹¹`

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## Problem 2: `(9 × 10⁵) + (5 × 10⁵)`

Same exponent → add coefficients: `9 + 5 = 14`
→ `14 × 10⁵`

Convert to proper scientific notation:
`14 × 10⁵ = 1.4 × 10⁶`

✔ Answer: `1.4 × 10⁶`

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## Problem 3: `(3 × 10⁸) − (7 × 10⁷)`

Exponents differ → rewrite one to match.

Convert `7 × 10⁷` to exponent 8:
`7 × 10⁷ = 0.7 × 10⁸`

Now subtract:
`(3 × 10⁸) − (0.7 × 10⁸) = (3 − 0.7) × 10⁸ = 2.3 × 10⁸`

✔ Answer: `2.3 × 10⁸`

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## Problem 4: `(5 × 10⁸) / (4 × 10³)`

Divide coefficients: `5 ÷ 4 = 1.25`
Subtract exponents: `8 − 3 = 5`
→ `1.25 × 10⁵`

Already in correct form.

✔ Answer: `1.25 × 10⁵`

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## Problem 5: `(2 × 10⁴) / (8 × 10²)`

Divide coefficients: `2 ÷ 8 = 0.25`
Subtract exponents: `4 − 2 = 2`
→ `0.25 × 10²`

Convert to scientific notation:
`0.25 × 10² = 2.5 × 10¹`

✔ Answer: `2.5 × 10¹`

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## Problem 6: `(2 × 10⁷) − (4 × 10⁷)`

Same exponent → subtract coefficients: `2 − 4 = -2`
→ `-2 × 10⁷`

Already in correct form (coefficient between -10 and -1? Wait — scientific notation allows negative coefficients as long as absolute value is ≥1 and <10).

✔ Answer: `-2 × 10⁷`

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## Problem 7: `(6 × 10⁴) + (1 × 10⁵)`

Different exponents → convert `1 × 10⁵` to exponent 4:
`1 × 10⁵ = 10 × 10⁴`

Now add:
`(6 × 10⁴) + (10 × 10⁴) = 16 × 10⁴`

Convert: `16 × 10⁴ = 1.6 × 10⁵`

✔ Answer: `1.6 × 10⁵`

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## Problem 8: `(8 × 10⁵)(6 × 10⁷)`

Multiply coefficients: `8 × 6 = 48`
Add exponents: `5 + 7 = 12`
→ `48 × 10¹²`

Convert: `48 × 10¹² = 4.8 × 10¹³`

✔ Answer: `4.8 × 10¹³`

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## Problem 9: `(4 × 10⁹)(9 × 10⁵)`

Multiply coefficients: `4 × 9 = 36`
Add exponents: `9 + 5 = 14`
→ `36 × 10¹⁴`

Convert: `36 × 10¹⁴ = 3.6 × 10¹⁵`

✔ Answer: `3.6 × 10¹⁵`

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## Problem 10: `(3 × 10⁵) + (7 × 10⁵)`

Same exponent → add coefficients: `3 + 7 = 10`
→ `10 × 10⁵`

Convert: `10 × 10⁵ = 1 × 10⁶`

✔ Answer: `1 × 10⁶`

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## ✔ Final Answers:

1) `1.4 × 10¹¹`
2) `1.4 × 10⁶`
3) `2.3 × 10⁸`
4) `1.25 × 10⁵`
5) `2.5 × 10¹`
6) `-2 × 10⁷`
7) `1.6 × 10⁵`
8) `4.8 × 10¹³`
9) `3.6 × 10¹⁵`
10) `1 × 10⁶`

Let me know if you’d like a printable version or step-by-step breakdown for any specific problem!