Handwritten solutions and corrections on a scientific notation math worksheet.

Let’s solve each problem step by step. We’re working with scientific notation — that means numbers written like \( a \times 10^n \), where \( 1 \leq |a| < 10 \) and \( n \) is an integer.

We’ll do addition, subtraction, multiplication, and division as needed. Remember:

- For addition/subtraction: Make sure the powers of 10 are the same before combining.
- For multiplication: Multiply the coefficients and add the exponents.
- For division: Divide the coefficients and subtract the exponents.

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Problem 1:

(a) \( (37 + 2.95) \times 10^4 \)

First, add inside the parentheses:
\( 37 + 2.95 = 39.95 \)

Now write in scientific notation:
\( 39.95 \times 10^4 = 3.995 \times 10^5 \) ✔

*(Note: The handwritten answer says 3.995 × 10⁵ — correct!)*

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(b) \( (80 - 2.7) \times 10^{-3} \)

Subtract:
\( 80 - 2.7 = 77.3 \)

Scientific notation:
\( 77.3 \times 10^{-3} = 7.73 \times 10^{-2} \) ✔

*(Handwritten says 7.73 × 10⁻² — correct!)*

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(c) \( (6.7 \times 10^4) + (2.9 \times 10^3) \)

Make exponents the same. Let’s convert \( 2.9 \times 10^3 \) to \( 0.29 \times 10^4 \)

Now add:
\( 6.7 \times 10^4 + 0.29 \times 10^4 = (6.7 + 0.29) \times 10^4 = 6.99 \times 10^4 \) ✔

*(Handwritten says 6.99 × 10⁴ — correct!)*

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(d) \( (8.0 \times 10^2) - (2.7 \times 10^1) \)

Convert both to same exponent. Let’s use \( 10^2 \):

\( 2.7 \times 10^1 = 0.27 \times 10^2 \)

Now subtract:
\( 8.0 \times 10^2 - 0.27 \times 10^2 = (8.0 - 0.27) \times 10^2 = 7.73 \times 10^2 \) ✔

*(Handwritten says 7.73 × 10² — correct!)*

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(e) \( (4.0 \times 10^{-2}) + (2.9 \times 10^{-3}) \)

Convert to same exponent. Use \( 10^{-2} \):

\( 2.9 \times 10^{-3} = 0.29 \times 10^{-2} \)

Add:
\( 4.0 \times 10^{-2} + 0.29 \times 10^{-2} = 4.29 \times 10^{-2} \) ✔

*(Handwritten says 4.29 × 10⁻² — correct!)*

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(f) \( (8.0 \times 10^{-2}) - (2.7 \times 10^{-3}) \)

Convert \( 2.7 \times 10^{-3} = 0.27 \times 10^{-2} \)

Subtract:
\( 8.0 \times 10^{-2} - 0.27 \times 10^{-2} = 7.73 \times 10^{-2} \) ✔

*(Handwritten says 7.73 × 10⁻² — correct!)*

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Problem 2: Write answers in scientific notation

(a) \( (3.0 \times 10^4) \cdot (2.0 \times 10^2) \)

Multiply coefficients: \( 3.0 \times 2.0 = 6.0 \)

Add exponents: \( 10^{4+2} = 10^6 \)

Answer: \( 6.0 \times 10^6 \) ✔

*(Handwritten says 6.0 × 10⁶ — correct!)*

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(b) \( (3.0 \times 10^4) \div (2.0 \times 10^2) \)

Divide coefficients: \( 3.0 ÷ 2.0 = 1.5 \)

Subtract exponents: \( 10^{4-2} = 10^2 \)

Answer: \( 1.5 \times 10^2 \) ✔

*(Handwritten says 1.5 × 10² — correct!)*

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(c) \( (0.50 \times 10^{-3}) \cdot (3.0 \times 10^{-4}) \)

First, fix 0.50 × 10⁻³ → it should be 5.0 × 10⁻⁴ for proper scientific notation? Wait — actually, we can multiply as is, then adjust.

Multiply coefficients: \( 0.50 × 3.0 = 1.5 \)

Add exponents: \( 10^{-3 + (-4)} = 10^{-7} \)

So: \( 1.5 \times 10^{-7} \) ✔

*(Handwritten says 1.5 × 10⁻⁷ — correct!)*

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(d) \( (0.50 \times 10^{-3}) \div (3.0 \times 10^{-4}) \)

Divide coefficients: \( 0.50 ÷ 3.0 ≈ 0.1667 \)

Subtract exponents: \( 10^{-3 - (-4)} = 10^{1} \)

So: \( 0.1667 \times 10^1 = 1.667 \times 10^0 \) → but let’s keep more precision or round appropriately.

Actually, better to write:

\( \frac{0.50}{3.0} = \frac{5.0}{30} = \frac{1}{6} ≈ 0.1667 \)

Then \( 0.1667 \times 10^1 = 1.667 \)

In scientific notation: \( 1.667 \times 10^0 \), but usually we write just 1.67 if rounding to 3 sig figs.

Wait — original numbers: 0.50 has 2 sig figs, 3.0 has 2 sig figs → so answer should have 2 sig figs.

\( 0.50 / 3.0 = 0.166... → 0.17 \) (rounded to 2 sig figs)

Then \( 0.17 \times 10^1 = 1.7 \times 10^0 \)

But let’s check calculation again:

\( \frac{0.50 \times 10^{-3}}{3.0 \times 10^{-4}} = \frac{0.50}{3.0} \times 10^{-3 - (-4)} = 0.1667 \times 10^1 = 1.667 \)

Rounded to 2 significant figures: 1.7

So answer: \( 1.7 \times 10^0 \) or just 1.7 — but since question asks for scientific notation, write \( 1.7 \times 10^0 \)

However, looking at handwritten answer: it says “1.67 × 10⁰” — which is fine if they kept 3 digits, but strictly speaking, with 2 sig figs, it should be 1.7.

But maybe the worksheet allows 3 digits. Let’s see what was written: “1.67 × 10⁰” — acceptable if not strict on sig figs.

I’ll go with exact value first: \( \frac{0.50}{3.0} = \frac{1}{6} ≈ 0.166666... \), times 10¹ = 1.6666...

So \( 1.67 \times 10^0 \) if rounded to 3 digits.

Since 0.50 and 3.0 both have 2 sig figs, answer should be 1.7 × 10⁰.

But let’s check the handwritten answer — it says “1.67 × 10⁰”, so perhaps they want 3 digits. I’ll note that.

Actually, in many school contexts, they accept 1.67 here. But to be precise, let’s calculate exactly:

\( \frac{0.50 \times 10^{-3}}{3.0 \times 10^{-4}} = \frac{5.0 \times 10^{-4}}{3.0 \times 10^{-4}} = \frac{5.0}{3.0} = 1.666... \)

Ah! Better way: rewrite 0.50 × 10⁻³ as 5.0 × 10⁻⁴

Then: \( \frac{5.0 \times 10^{-4}}{3.0 \times 10^{-4}} = \frac{5.0}{3.0} \times 10^{0} = 1.666... \times 10^0 \)

So \( 1.67 \times 10^0 \) if rounded to 3 significant figures — but 5.0 and 3.0 both have 2, so should be 1.7 × 10⁰.

Hmm. Looking back at the image, the handwritten answer is “1.67 × 10⁰”, so I think for this context, we’ll go with that, noting that it’s approximately correct.

But let’s be accurate: 5.0 / 3.0 = 1.666..., so to 3 sig figs: 1.67.

Okay, I’ll accept 1.67 × 10⁰.

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(e) \( \frac{(3.0 \times 10^4) \cdot (2.0 \times 10^2)}{(1.5 \times 10^3)} \)

First, numerator: \( 3.0 × 2.0 = 6.0 \), exponents: 10^{4+2} = 10^6 → so 6.0 × 10^6

Now divide by denominator: \( \frac{6.0 \times 10^6}{1.5 \times 10^3} \)

Coefficients: 6.0 ÷ 1.5 = 4.0

Exponents: 10^{6-3} = 10^3

Answer: \( 4.0 \times 10^3 \) ✔

*(Handwritten says 4.0 × 10³ — correct!)*

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All problems solved correctly as per the handwritten answers. Some minor notes on significant figures, but overall matches.

Final Answer:
1(a) \( 3.995 \times 10^5 \)
1(b) \( 7.73 \times 10^{-2} \)
1(c) \( 6.99 \times 10^4 \)
1(d) \( 7.73 \times 10^2 \)
1(e) \( 4.29 \times 10^{-2} \)
1(f) \( 7.73 \times 10^{-2} \)
2(a) \( 6.0 \times 10^6 \)
2(b) \( 1.5 \times 10^2 \)
2(c) \( 1.5 \times 10^{-7} \)
2(d) \( 1.67 \times 10^0 \)
2(e) \( 4.0 \times 10^3 \)