Math worksheet for simplifying operations with scientific notation.

Let’s solve each problem step by step. We’ll use the rules of scientific notation:

- When multiplying: multiply the coefficients and add the exponents.
- When dividing: divide the coefficients and subtract the exponents.
- When raising to a power: raise the coefficient to that power and multiply the exponent by that power.
- Always write the final answer in proper scientific notation (coefficient between 1 and 10).

---

Problem 1:
(1.08 × 10⁻⁵)(9.3 × 10⁻²)
Multiply coefficients: 1.08 × 9.3 = 10.044
Add exponents: -5 + (-2) = -7
So we have: 10.044 × 10⁻⁷ → but this is not proper scientific notation (10.044 > 10).
Adjust: 10.044 × 10⁻⁷ = 1.0044 × 10⁻⁶

✔ Final Answer for #1: 1.0044 × 10⁻⁶

---

Problem 2:
(2 × 10⁻⁴)(8.1 × 10⁻¹)
Coefficients: 2 × 8.1 = 16.2
Exponents: -4 + (-1) = -5
→ 16.2 × 10⁻⁵ → adjust: 1.62 × 10⁻⁴

✔ Final Answer for #2: 1.62 × 10⁻⁴

---

Problem 3:
(2.32 × 10⁻⁴)(4 × 10⁻⁵)
Coefficients: 2.32 × 4 = 9.28
Exponents: -4 + (-5) = -9
→ Already good: 9.28 × 10⁻⁹

✔ Final Answer for #3: 9.28 × 10⁻⁹

---

Problem 4:
(3.48 × 10³)(9.8 × 10⁴)
Coefficients: 3.48 × 9.8 = let’s calculate:
3.48 × 10 = 34.8, minus 3.48 × 0.2 = 0.696 → 34.8 - 0.696? Wait — no!
Actually: 3.48 × 9.8 = 3.48 × (10 - 0.2) = 34.8 - 0.696 = 34.104
Wait — better to do direct multiplication:
3.48 × 9.8
= 3.48 × 9 + 3.48 × 0.8
= 31.32 + 2.784 = 34.104
Exponents: 3 + 4 = 7
→ 34.104 × 10⁷ → adjust: 3.4104 × 10⁸

✔ Final Answer for #4: 3.4104 × 10⁸

---

Problem 5:
(7.1 × 10⁻⁵)(6.7 × 10⁻⁸)
Coefficients: 7.1 × 6.7
7 × 6.7 = 46.9, 0.1 × 6.7 = 0.67 → total 47.57
Exponents: -5 + (-8) = -13
→ 47.57 × 10⁻¹³ → adjust: 4.757 × 10⁻¹²

✔ Final Answer for #5: 4.757 × 10⁻¹²

---

Problem 6:
(6 × 10³)(9.91 × 10⁰)
Note: 10⁰ = 1
Coefficients: 6 × 9.91 = 59.46
Exponents: 3 + 0 = 3
→ 59.46 × 10³ → adjust: 5.946 × 10⁴

✔ Final Answer for #6: 5.946 × 10⁴

---

Problem 7:
(7.1 × 10⁶) / (8.2 × 10¹)
Divide coefficients: 7.1 ÷ 8.2 ≈ ?
7.1 ÷ 8.2 = 71/82 ≈ 0.86585...
Subtract exponents: 6 - 1 = 5
→ 0.86585 × 10⁵ → adjust: 8.6585 × 10⁴
But let’s keep more precision or round appropriately? The problem doesn’t specify rounding, so we can leave as is or round to reasonable digits. Since inputs have 2 sig figs mostly, but 7.1 and 8.2 both have 2, so maybe 2 sig figs? But let’s check original numbers — actually, 7.1 has 2, 8.2 has 2 → so answer should have 2 sig figs? But in math class sometimes they want exact. Let’s compute exactly.

Actually, 7.1 / 8.2 = 71/82 = approximately 0.8658536585
× 10⁵ → 8.658536585 × 10⁴ → if we round to 3 sig figs (since 7.1 and 8.2 are 2, but maybe accept 3), but let’s see other problems — many have 3 digits. I’ll go with 3 sig figs unless specified.

But wait — problem says “write each answer in scientific notation” — doesn’t specify sig figs. So perhaps keep full precision from calculation? But 7.1 and 8.2 are given with 2 decimal places but only 2 sig figs? Actually 7.1 has 2 sig figs, 8.2 has 2 → so quotient should have 2 sig figs.

So 0.87 × 10⁵ = 8.7 × 10⁴

But let me double-check: 7.1 ÷ 8.2 = 0.86585… → rounded to 2 sig figs is 0.87 → then 0.87 × 10⁵ = 8.7 × 10⁴

✔ Final Answer for #7: 8.7 × 10⁴ (rounded to 2 significant figures)

Wait — but looking at other problems, like #1 had 1.08 and 9.3 — 3 and 2 sig figs, and we kept 5 digits? Hmm. Maybe the worksheet expects exact computation without rounding unless necessary. Let me recalculate without rounding early.

Actually, in most algebra classes, they expect you to compute the number and write it in sci notation, even if it has many digits. For example, problem 1 we wrote 1.0044 × 10⁻⁶.

So for consistency, let’s not round unless the result is messy. But 7.1 / 8.2 is repeating? Let’s do fraction: 71/82 × 10⁵ = (71 × 10⁵)/82

Calculate numerically: 7.1e6 / 8.2e1 = 7100000 / 82 = let's divide:

82 × 86585 = ? Better use calculator in mind: 7100000 ÷ 82.

82 × 86000 = 82×86×1000 = (80×86=6880, 2×86=172, total 7052) ×1000 = 7,052,000
Subtract from 7,100,000: 48,000 left
82 × 585 = 82×500=41,000; 82×85=6970; total 41,000+6,970=47,970
Close, remainder 30 → so approximately 86585.365...

So 86585.365... which is 8.6585365... × 10⁴

Since the problem didn't specify rounding, and others were left with multiple decimals, I'll write it as 8.6585 × 10⁴? But that's arbitrary. Looking back at problem 1: 1.08 * 9.3 = 10.044 — they probably expect exact product.

Similarly here, 7.1 / 8.2 = 71/82, which is exact fraction, but in decimal it's repeating. Perhaps leave as decimal with several digits.

But to match style, let's do what we did in #1: compute exactly and write in sci notation.

7.1 × 10⁶ / 8.2 × 10¹ = (7.1 / 8.2) × 10^(6-1) = (71/82) × 10⁵

71 ÷ 82 = 0.8658536585365854...

So 0.8658536585365854 × 10⁵ = 8.658536585365854 × 10⁴

We can round to 4 or 5 digits? But in #1 we had 1.0044 which is 5 digits. Let's use 4 digits after decimal for consistency? Or perhaps the worksheet expects simplified form.

I think for school level, they might expect us to compute and write the number as is. Let me check online or standard practice — but since I can't, I'll assume we keep it as calculated.

But to be safe, let's look at problem 8: 5.4e-1 / 3.4e3 — similar.

Perhaps for now, I'll write it as 8.66 × 10⁴ if rounding to 3 sig figs, but 7.1 and 8.2 have 2 sig figs, so 8.7 × 10⁴.

I recall that in multiplication/division, the number with the least number of significant figures determines the answer. Here both have 2, so answer should have 2.

So 7.1 / 8.2 = 0.87 (2 sig figs) → 0.87 × 10^5 = 8.7 × 10^4

Yes, that makes sense.

✔ Final Answer for #7: 8.7 × 10⁴

---

Problem 8:
(5.4 × 10⁻¹) / (3.4 × 10³)
Coefficients: 5.4 ÷ 3.4 ≈ 1.588235...
Exponents: -1 - 3 = -4
→ 1.588235 × 10⁻⁴
Sig figs: 5.4 and 3.4 both have 2 sig figs → so round to 2 sig figs: 1.6 × 10⁻⁴

✔ Final Answer for #8: 1.6 × 10⁻⁴

---

Problem 9:
(4 × 10⁴) / (3.63 × 10⁻⁴)
Coefficients: 4 ÷ 3.63 ≈ 1.101928...
Exponents: 4 - (-4) = 8
→ 1.101928 × 10⁸
Sig figs: 4 has 1 sig fig? Wait, 4 could be considered as 1 sig fig, but in context, it might be exact. 3.63 has 3 sig figs. Usually, if it's written as "4", it might be ambiguous, but in scientific notation problems, often treated as exact or with implied precision. To be safe, since 4 is integer, perhaps keep as is. But let's see — 4.0 would be 2 sig figs, but it's just "4". In many textbooks, they treat single digit integers in such contexts as having infinite sig figs or match the other number. But here, 3.63 has 3, so perhaps answer with 3 sig figs.

4 ÷ 3.63 = 1.101928... → to 3 sig figs: 1.10 × 10⁸

But 1.10 has 3 sig figs.

Alternatively, if 4 is exact, then limited by 3.63's 3 sig figs.

I think 1.10 × 10⁸ is fine.

Compute: 4 / 3.63 = 400 / 363 ≈ 1.101928 → yes.

✔ Final Answer for #9: 1.10 × 10⁸

---

Problem 10:
(9 × 10⁻⁵) / (9.24 × 10⁻⁶)
Coefficients: 9 ÷ 9.24 ≈ 0.97402597...
Exponents: -5 - (-6) = 1
→ 0.97402597 × 10¹ = 9.7402597
Now, sig figs: 9 has 1 sig fig, 9.24 has 3 → so answer should have 1 sig fig? That would be 10, but that seems odd.

9 could be considered as 9.0, but it's written as "9". In scientific notation, sometimes it's assumed to be precise. To avoid under-rounding, perhaps keep 3 digits as per 9.24.

But strictly, 9 has 1 sig fig, so 9 / 9.24 ≈ 1 (since 0.97 rounds to 1 with 1 sig fig) → 1 × 10¹ = 10

But that loses information. Perhaps in this context, since it's math not science, they want exact calculation.

Let me calculate: 9e-5 / 9.24e-6 = 9/9.24 * 10^{1} = (900/924) * 10 = simplify 900÷12=75, 924÷12=77? 900 and 924 divided by 12: 900÷12=75, 924÷12=77, so 75/77 * 10 ≈ 0.97402597 * 10 = 9.7402597

So approximately 9.74 if rounded to 3 sig figs.

Given that other answers have multiple digits, I'll go with 9.74 × 10⁰, but usually we write without 10⁰.

9.74 × 10⁰ is just 9.74, but in scientific notation, it's acceptable, but typically for numbers between 1 and 10, we don't write ×10⁰.

The problem says "scientific notation", which requires a coefficient between 1 and 10 times a power of 10. 9.74 is between 1 and 10, so it should be 9.74 × 10⁰, but often omitted. However, to be precise, we should include it.

But in practice, for numbers like this, they might expect 9.74.

Looking at problem 6, they had 5.946 × 10⁴, so for 9.74, it should be 9.74 × 10⁰.

But let's see the exponent: we have 0.974 × 10^1 = 9.74 × 10^0, yes.

And sig figs: if we take 9 as 1 sig fig, it should be 10, but that seems too rough. Perhaps the "9" is meant to be exact. I think for this worksheet, they expect the calculation as is.

I'll put 9.74 × 10⁰, but to make it neat, perhaps 9.74.

But the instruction is "scientific notation", so must have ×10^n.

So 9.74 × 10⁰

But let's confirm the calculation: 9e-5 / 9.24e-6 = 9/9.24 * 10^{-5 - (-6)} = 0.97402597 * 10^1 = 9.7402597, so 9.74 × 10^0 if rounded to 3 sig figs.

Since 9.24 has 3, and 9 might be considered as 9.00, I'll use 3 sig figs.

✔ Final Answer for #10: 9.74 × 10⁰

But typically, we write it as 9.74, but to follow instructions, I'll keep the ×10⁰.

Some might argue that 9.74 is already in scientific notation with exponent 0 implied, but technically, scientific notation includes the power of 10.

To be safe, I'll write it as 9.74 × 10⁰.

---

Problem 11:
(8.42 × 10³) / (5 × 10²)
Coefficients: 8.42 ÷ 5 = 1.684
Exponents: 3 - 2 = 1
→ 1.684 × 10¹
Sig figs: 8.42 has 3, 5 has 1 → so should have 1 sig fig? 2 × 10¹ = 20

But again, 5 might be exact. In many cases, if it's "5", it could be considered as 5.0, but here it's written as "5". To match the style, perhaps keep as 1.684 × 10¹, but let's see.

If we follow sig fig rules strictly, 5 has 1 sig fig, so answer should be 2 × 10¹.

But in the context of this worksheet, since other problems have detailed decimals, perhaps they want the exact value.

I think for consistency with problem 9, where we used 3 sig figs for 3.63, here 5 might be limiting.

But let's calculate: 8.42e3 / 5e2 = 8420 / 500 = 16.84 / 10? 8420 ÷ 500 = 16.84, which is 1.684 × 10^1.

Perhaps write as 1.684 × 10^1, and since the problem didn't specify, leave it.

But to be safe, I'll note that if sig figs are required, it would be 2 × 10^1, but I think for this, they want the calculation.

Looking at problem 12: 8.9e6 / 8.4e2 — similar.

I'll go with 1.684 × 10^1 for now.

✔ Final Answer for #11: 1.684 × 10¹

---

Problem 12:
(8.9 × 10⁶) / (8.4 × 10²)
Coefficients: 8.9 ÷ 8.4 ≈ 1.0595238...
Exponents: 6 - 2 = 4
→ 1.0595238 × 10⁴
Sig figs: both have 2 sig figs → so round to 2 sig figs: 1.1 × 10⁴

8.9 / 8.4 = 89/84 ≈ 1.0595, which rounds to 1.1 with 2 sig figs.

✔ Final Answer for #12: 1.1 × 10⁴

---

Problem 13:
(8.9 × 10³)^4
Raise coefficient to 4th power: 8.9^4
First, 8.9^2 = 79.21
Then 79.21^2 = ?
79.21 × 79.21
Calculate: 80×80=6400, but more accurately:
(80 - 0.79)^2 = 80^2 - 2*80*0.79 + (0.79)^2 = 6400 - 126.4 + 0.6241 = 6400 - 126.4 = 6273.6, +0.6241=6274.2241
Better: 79.21 × 79.21
Or: 7921/100 * 7921/100 = (7921^2)/10000
But easier: 79.21 × 79.21
Let me compute:
79.21 × 80 = 6336.8
Minus 79.21 × 0.79 = ? First, 79.21 × 0.8 = 63.368, minus 79.21 × 0.01 = 0.7921, so 63.368 - 0.7921 = 62.5759? No:
Actually, 79.21 × 0.79 = 79.21 × (0.8 - 0.01) = 79.21×0.8 - 79.21×0.01 = 63.368 - 0.7921 = 62.5759
But since we're doing 79.21 × 79.21, not related.

Standard multiplication:
79.21
× 79.21
First, 7921 × 7921 / 10000, but perhaps decimal.

79.21 × 79.21 = (79 + 0.21)^2 = 79^2 + 2*79*0.21 + 0.21^2 = 6241 + 33.18 + 0.0441 = 6274.2241

Yes.

So 8.9^4 = (8.9^2)^2 = 79.21^2 = 6274.2241

Now exponents: (10^3)^4 = 10^{12}

So overall: 6274.2241 × 10^{12}

Now convert to scientific notation: 6.2742241 × 10^3 × 10^{12} = 6.2742241 × 10^{15}

Sig figs: 8.9 has 2 sig figs, so answer should have 2 sig figs: 6.3 × 10^{15}

✔ Final Answer for #13: 6.3 × 10¹⁵

---

Problem 14:
(4 × 10⁻⁵)^{-6}
First, apply the exponent to both parts:
4^{-6} × (10^{-5})^{-6} = 4^{-6} × 10^{30} (since (-5)*(-6)=30)

4^{-6} = 1 / 4^6
4^2=16, 4^4=(16)^2=256, 4^6=256×16=4096
So 1/4096 ≈ 0.000244140625

So 0.000244140625 × 10^{30} = 2.44140625 × 10^{-4} × 10^{30} = 2.44140625 × 10^{26}

Sig figs: 4 has 1 sig fig, so answer should have 1 sig fig: 2 × 10^{26}

But let's confirm: 4^{-6} = 1/4096 ≈ 0.00024414, which is 2.4414 × 10^{-4}, times 10^{30} = 2.4414 × 10^{26}

With 1 sig fig, it's 2 × 10^{26}

✔ Final Answer for #14: 2 × 10²⁶

---

Now, compiling all final answers:

1) 1.0044 × 10⁻⁶
2) 1.62 × 10⁻⁴
3) 9.28 × 10⁻⁹
4) 3.4104 × 10⁸
5) 4.757 × 10⁻¹²
6) 5.946 × 10⁴
7) 8.7 × 10⁴
8) 1.6 × 10⁻⁴
9) 1.10 × 10⁸
10) 9.74 × 10⁰
11) 1.684 × 10¹
12) 1.1 × 10⁴
13) 6.3 × 10¹⁵
14) 2 × 10²⁶

For problem 10, 9.74 × 10⁰ is correct, but sometimes written as 9.74, but to be consistent with scientific notation format, I'll keep it.

Similarly, for 11, 1.684 × 10¹ is 16.84, but in sci notation it's fine.

I think this is complete.

Final Answer:
1) \( 1.0044 \times 10^{-6} \)
2) \( 1.62 \times 10^{-4} \)
3) \( 9.28 \times 10^{-9} \)
4) \( 3.4104 \times 10^{8} \)
5) \( 4.757 \times 10^{-12} \)
6) \( 5.946 \times 10^{4} \)
7) \( 8.7 \times 10^{4} \)
8) \( 1.6 \times 10^{-4} \)
9) \( 1.10 \times 10^{8} \)
10) \( 9.74 \times 10^{0} \)
11) \( 1.684 \times 10^{1} \)
12) \( 1.1 \times 10^{4} \)
13) \( 6.3 \times 10^{15} \)
14) \( 2 \times 10^{26} \)