Scientific notation practice problems for multiplication and division.

Let's solve each of these problems step by step. These involve scientific notation and operations (multiplication and division). The key rules are:

Rules for Scientific Notation:

- Multiplication:
$(a \times 10^m)(b \times 10^n) = (a \times b) \times 10^{m+n}$
- Division:
$\frac{a \times 10^m}{b \times 10^n} = \left(\frac{a}{b}\right) \times 10^{m-n}$

We'll apply these rules to each problem.

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1) $(8.18 \times 10^{-6})(1.15 \times 10^{-5})$

Multiply coefficients:
$8.18 \times 1.15 = 9.407$

Add exponents:
$-6 + (-5) = -11$

So:
$9.407 \times 10^{-11}$

✔ Answer: $9.407 \times 10^{-11}$

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2) $(5.8 \times 10^{-6})(2 \times 10^4)$

$5.8 \times 2 = 11.6$

Exponent: $-6 + 4 = -2$

So: $11.6 \times 10^{-2}$ → But we need to write in proper scientific notation (coefficient < 10)

$11.6 = 1.16 \times 10^1$, so:

$1.16 \times 10^1 \times 10^{-2} = 1.16 \times 10^{-1}$

✔ Answer: $1.16 \times 10^{-1}$

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3) $(0.8 \times 10^4)(1.28 \times 10^6)$

First, convert $0.8 \times 10^4$ to standard scientific notation:
$0.8 = 8 \times 10^{-1}$, so:
$8 \times 10^{-1} \times 10^4 = 8 \times 10^3$

Now multiply:
$(8 \times 10^3)(1.28 \times 10^6) = (8 \times 1.28) \times 10^{3+6} = 10.24 \times 10^9$

Convert to proper form:
$10.24 = 1.024 \times 10^1$, so:
$1.024 \times 10^1 \times 10^9 = 1.024 \times 10^{10}$

✔ Answer: $1.024 \times 10^{10}$

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4) $(3.8 \times 10^{-6})(2.37 \times 10^{-3})$

$3.8 \times 2.37 = 9.006$

Exponent: $-6 + (-3) = -9$

So: $9.006 \times 10^{-9}$

✔ Answer: $9.006 \times 10^{-9}$

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5) $(1.9 \times 10^{-3})(2 \times 10^4)$

$1.9 \times 2 = 3.8$

Exponent: $-3 + 4 = 1$

So: $3.8 \times 10^1 = 38$

But let’s keep it in scientific notation: $3.8 \times 10^1$

✔ Answer: $3.8 \times 10^1$

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6) $(9.2 \times 10^5)(4 \times 10^{-3})$

$9.2 \times 4 = 36.8$

Exponent: $5 + (-3) = 2$

So: $36.8 \times 10^2$

Convert: $36.8 = 3.68 \times 10^1$, so:

$3.68 \times 10^1 \times 10^2 = 3.68 \times 10^3$

✔ Answer: $3.68 \times 10^3$

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7) $\frac{7.8 \times 10^4}{8 \times 10^1}$

Divide coefficients: $\frac{7.8}{8} = 0.975$

Subtract exponents: $4 - 1 = 3$

So: $0.975 \times 10^3$

Convert: $0.975 = 9.75 \times 10^{-1}$, so:

$9.75 \times 10^{-1} \times 10^3 = 9.75 \times 10^2$

✔ Answer: $9.75 \times 10^2$

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8) $\frac{5.3 \times 10^3}{7.65 \times 10^5}$

$\frac{5.3}{7.65} \approx 0.6923$

Exponent: $3 - 5 = -2$

So: $0.6923 \times 10^{-2}$

Convert: $0.6923 = 6.923 \times 10^{-1}$

So: $6.923 \times 10^{-1} \times 10^{-2} = 6.923 \times 10^{-3}$

✔ Answer: $6.923 \times 10^{-3}$

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9) $\frac{4.6 \times 10^2}{5.01 \times 10^{-3}}$

$\frac{4.6}{5.01} \approx 0.91816$

Exponent: $2 - (-3) = 5$

So: $0.91816 \times 10^5$

Convert: $0.91816 = 9.1816 \times 10^{-1}$

So: $9.1816 \times 10^{-1} \times 10^5 = 9.1816 \times 10^4$

✔ Answer: $9.18 \times 10^4$ (rounded to 3 significant figures)

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10) $\frac{7.6 \times 10^0}{5.4 \times 10^{-6}}$

$10^0 = 1$, so numerator is $7.6$

$\frac{7.6}{5.4} \approx 1.4074$

Exponent: $0 - (-6) = 6$

So: $1.4074 \times 10^6$

✔ Answer: $1.41 \times 10^6$ (rounded to 3 sig figs)

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11) $\frac{5.5 \times 10^{-1}}{5.3 \times 10^2}$

$\frac{5.5}{5.3} \approx 1.0377$

Exponent: $-1 - 2 = -3$

So: $1.0377 \times 10^{-3}$

✔ Answer: $1.04 \times 10^{-3}$ (rounded to 3 sig figs)

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12) $\frac{2.04 \times 10^{-1}}{2 \times 10^{-2}}$

$\frac{2.04}{2} = 1.02$

Exponent: $-1 - (-2) = 1$

So: $1.02 \times 10^1 = 10.2$

✔ Answer: $1.02 \times 10^1$

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✔ Final Answers:

| Problem | Answer |
|--------|--------|
| 1) | $9.407 \times 10^{-11}$ |
| 2) | $1.16 \times 10^{-1}$ |
| 3) | $1.024 \times 10^{10}$ |
| 4) | $9.006 \times 10^{-9}$ |
| 5) | $3.8 \times 10^1$ |
| 6) | $3.68 \times 10^3$ |
| 7) | $9.75 \times 10^2$ |
| 8) | $6.923 \times 10^{-3}$ |
| 9) | $9.18 \times 10^4$ |
| 10) | $1.41 \times 10^6$ |
| 11) | $1.04 \times 10^{-3}$ |
| 12) | $1.02 \times 10^1$ |

Let me know if you'd like the answers rounded differently or explained further!