Examples of multiplication and division operations using scientific notation.

Let’s solve each problem one by one.

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Left Column: Multiplication

When multiplying numbers in scientific notation:
- Multiply the decimal parts (the numbers before ×10^something)
- Add the exponents of 10

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Problem 1:
(2.156 × 10²)(5.034 × 10⁵)

Step 1: Multiply decimals → 2.156 × 5.034
Let’s calculate that:

2.156 × 5 = 10.78
2.156 × 0.034 = let’s compute:
2.156 × 0.03 = 0.06468
2.156 × 0.004 = 0.008624
→ Total = 0.06468 + 0.008624 = 0.073304
So total = 10.78 + 0.073304 = 10.853304

Wait — better to do direct multiplication:

2.156 × 5.034
= (2 + 0.156) × (5 + 0.034)
But easier with calculator-style:

Actually, let’s just multiply directly:

2.156 × 5.034
First: 2156 × 5034 = ? Too big — use approximation or precise step:

Alternatively, accept we can compute:

2.156 × 5.034 ≈ 10.853304 (we’ll keep more digits for accuracy)

Actually, let me compute precisely:

2.156 × 5.034:

Break it down:

2.156 × 5 = 10.78
2.156 × 0.03 = 0.06468
2.156 × 0.004 = 0.008624
Add them: 10.78 + 0.06468 = 10.84468 + 0.008624 = 10.853304

Now add exponents: 10² × 10⁵ = 10^(2+5) = 10⁷

So result = 10.853304 × 10⁷

But scientific notation requires one digit before decimal → so rewrite as:

1.0853304 × 10⁸

✔ Final Answer for Problem 1: 1.0853304 × 10⁸

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Problem 2:
(6.74 × 10⁻⁴)(3.441 × 10⁵)

Multiply decimals: 6.74 × 3.441

Compute:

6.74 × 3 = 20.22
6.74 × 0.4 = 2.696
6.74 × 0.04 = 0.2696
6.74 × 0.001 = 0.00674

Better: 6.74 × 3.441

Do 674 × 3441 / 100000 (since both have 2 and 3 decimal places? Wait — 6.74 has 2 decimals, 3.441 has 3 → total 5 decimals)

Actually, just multiply:

6.74 × 3.441 = ?

Use:

6.74 × 3 = 20.22
6.74 × 0.4 = 2.696 → subtotal 22.916
6.74 × 0.04 = 0.2696 → subtotal 23.1856
6.74 × 0.001 = 0.00674 → total = 23.19234

Check: 6.74 × 3.441

Or use: 6.74 * 3.441 = let's say approximately 23.19234

Exponents: 10⁻⁴ × 10⁵ = 10¹

So: 23.19234 × 10¹ = 2.319234 × 10²

✔ Final Answer for Problem 2: 2.319234 × 10²

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Problem 3:
(5.07 × 10⁻²)(4.103 × 10⁻⁷)

Multiply decimals: 5.07 × 4.103

Compute:

5 × 4.103 = 20.515
0.07 × 4.103 = 0.28721
Total = 20.515 + 0.28721 = 20.80221

Exponents: 10⁻² × 10⁻⁷ = 10⁻⁹

So: 20.80221 × 10⁻⁹ = 2.080221 × 10⁻⁸

✔ Final Answer for Problem 3: 2.080221 × 10⁻⁸

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Right Column: Division

When dividing numbers in scientific notation:
- Divide the decimal parts
- Subtract the exponent of the divisor from the dividend

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Problem 4:
(7.126 × 10⁸) ÷ (3.37 × 10⁶)

Divide decimals: 7.126 ÷ 3.37

Calculate:

3.37 × 2 = 6.74 → subtract from 7.126 → remainder 0.386
Bring down zero → 3.860
3.37 × 1 = 3.37 → subtract → 0.490
Bring down zero → 4.900
3.37 × 1 = 3.37 → subtract → 1.530
Bring down zero → 15.30
3.37 × 4 = 13.48 → subtract → 1.82
... This is getting messy.

Better: 7.126 ÷ 3.37 ≈ ?

Use division: 7126 ÷ 3370 (multiply numerator and denominator by 1000)

7126 ÷ 3370 ≈ 2.1145...

Let me compute:

3.37 × 2.114 = ?

3.37 × 2 = 6.74
3.37 × 0.1 = 0.337 → 7.077
3.37 × 0.014 = 3.37 × 0.01 = 0.0337; 3.37 × 0.004 = 0.01348 → total 0.04718
So 7.077 + 0.04718 = 7.12418 → very close to 7.126

Difference: 7.126 - 7.12418 = 0.00182 → so add about 0.00182 / 3.37 ≈ 0.00054 → so approx 2.11454

So ≈ 2.1145

Exponents: 10⁸ 10⁶ = 10^(8-6) = 10²

So: 2.1145 × 10²

More accurately, let’s do exact division:

7.126 ÷ 3.37 = ?

Write as 7126 / 3370 = simplify?

Divide numerator and denominator by 2: 3563 / 1685

Now divide 3563 ÷ 1685:

1685 × 2 = 3370 → subtract from 3563 → 193
So 2 + 193/1685 ≈ 2 + 0.1145 = 2.1145

Yes.

So final: 2.1145 × 10²

But let’s check with calculator precision if allowed — since this is educational, we can round reasonably.

Actually, 7.126 ÷ 3.37 = 2.114539... → so 2.1145 is fine.

✔ Final Answer for Problem 4: 2.1145 × 10²

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Problem 5:
(5.0451 × 10⁻⁵) ÷ (8.32 × 10³)

Divide decimals: 5.0451 ÷ 8.32

This will be less than 1.

Compute: 5.0451 ÷ 8.32

≈ ?

8.32 × 0.6 = 4.992
Subtract from 5.0451 → 0.0531
Now 0.0531 ÷ 8.32 ≈ 0.00638

So total ≈ 0.60638

More precisely:

5.0451 ÷ 8.32 = ?

Multiply numerator and denominator by 10000: 50451 ÷ 83200

Or: 50451 / 83200 = ?

Divide: 50451 ÷ 83200 = 0.606382...

So ≈ 0.60638

Exponents: 10⁻⁵ ÷ 10³ = 10⁻⁵⁻³ = 10⁻⁸

So: 0.60638 × 10⁻⁸ = 6.0638 × 10⁻⁹

✔ Final Answer for Problem 5: 6.0638 × 10⁻⁹

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Problem 6:
(8.231 × 10⁶) ÷ (7.56 × 10⁻³)

Divide decimals: 8.231 ÷ 7.56

Compute:

7.56 × 1 = 7.56 → subtract from 8.231 → 0.671
0.671 ÷ 7.56 ≈ 0.08875

So total ≈ 1.08875

More precisely:

8.231 ÷ 7.56 = ?

7.56 × 1.088 = 7.56 × 1 = 7.56; 7.56 × 0.08 = 0.6048; 7.56 × 0.008 = 0.06048 → total 7.56 + 0.6048 = 8.1648 + 0.06048 = 8.22528

Close to 8.231 → difference 0.00572 → 0.00572 / 7.56 ≈ 0.000756 → so total ≈ 1.088756

So ≈ 1.08876

Exponents: 10⁶ ÷ 10⁻³ = 10^(6 - (-3)) = 10⁹

So: 1.08876 × 10⁹

✔ Final Answer for Problem 6: 1.08876 × 10⁹

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Final Answers:

Left column (multiplication):
1. 1.0853304 × 10⁸
2. 2.319234 × 10²
3. 2.080221 × 10⁻⁸

Right column (division):
4. 2.1145 × 10²
5. 6.0638 × 10⁻⁹
6. 1.08876 × 10⁹

We’ll present them in order as they appear.

Final Answer:
1.0853304 × 10⁸, 2.319234 × 10², 2.080221 × 10⁻⁸, 2.1145 × 10², 6.0638 × 10⁻⁹, 1.08876 × 10⁹