Practice problems for multiplying and dividing numbers in scientific notation.

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: $3.8 \times 10^1$

✔ 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 to scientific notation

$36.8 = 3.68 \times 10^1$, 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 to scientific notation:
$0.975 = 9.75 \times 10^{-1}$, 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: $9.182 \times 10^4$ (rounded to 4 sig figs)

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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: $1.407 \times 10^6$ (rounded to 4 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: $1.02 \times 10^1$

✔ Answer: $1.02 \times 10^1$

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✔ Final Answers Summary:

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) $9.182 \times 10^4$
10) $1.407 \times 10^6$
11) $1.04 \times 10^{-3}$
12) $1.02 \times 10^1$

Let me know if you'd like them rounded differently or with more/less significant figures!