Math worksheet for simplifying operations with scientific notation, including multiplication, division, and exponentiation.

Let's solve each problem step by step, simplifying expressions in scientific notation. Remember the rules:

Rules for Scientific Notation:

1. Multiplication:
$(a \times 10^m)(b \times 10^n) = (a \cdot b) \times 10^{m+n}$

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

3. Powers:
$(a \times 10^m)^n = a^n \times 10^{m \cdot n}$

We will simplify each expression and write the final answer in proper scientific notation: $a \times 10^n$, where $1 \leq a < 10$.

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1) $(1.08 \times 10^{-5})(9.3 \times 10^{-7})$

Multiply coefficients:
$1.08 \times 9.3 = 9.984$

Add exponents:
$-5 + (-7) = -12$

So:
$9.984 \times 10^{-12}$

✔ Answer: $9.984 \times 10^{-12}$

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2) $(2 \times 10^{-2})(8.1 \times 10^{-1})$

Coefficients:
$2 \times 8.1 = 16.2$

Exponents:
$-2 + (-1) = -3$

So: $16.2 \times 10^{-3}$ → not in proper form (16.2 > 10)

Adjust:
$16.2 = 1.62 \times 10^1$, so:
$1.62 \times 10^1 \times 10^{-3} = 1.62 \times 10^{-2}$

✔ Answer: $1.62 \times 10^{-2}$

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3) $(2.32 \times 10^{-4})(4 \times 10^{-5})$

Coefficients:
$2.32 \times 4 = 9.28$

Exponents:
$-4 + (-5) = -9$

✔ Answer: $9.28 \times 10^{-9}$

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4) $(3.48 \times 10^3)(9.8 \times 10^6)$

Coefficients:
$3.48 \times 9.8 = 34.104$

Exponents:
$3 + 6 = 9$

So: $34.104 \times 10^9$ → adjust

$34.104 = 3.4104 \times 10^1$, so:
$3.4104 \times 10^1 \times 10^9 = 3.4104 \times 10^{10}$

✔ Answer: $3.4104 \times 10^{10}$

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5) $(7.1 \times 10^{-3})(6.7 \times 10^{-6})$

Coefficients:
$7.1 \times 6.7 = 47.57$

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

So: $47.57 \times 10^{-9}$ → adjust

$47.57 = 4.757 \times 10^1$, so:
$4.757 \times 10^1 \times 10^{-9} = 4.757 \times 10^{-8}$

✔ Answer: $4.757 \times 10^{-8}$

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6) $(6 \times 10^2)(9.91 \times 10^6)$

Coefficients:
$6 \times 9.91 = 59.46$

Exponents:
$2 + 6 = 8$

So: $59.46 \times 10^8$ → adjust

$59.46 = 5.946 \times 10^1$, so:
$5.946 \times 10^1 \times 10^8 = 5.946 \times 10^9$

✔ Answer: $5.946 \times 10^9$

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7) $\frac{7.1 \times 10^6}{8.2 \times 10^1}$

Divide coefficients:
$\frac{7.1}{8.2} \approx 0.86585$

Subtract exponents:
$6 - 1 = 5$

So: $0.86585 \times 10^5$ → not proper

Convert: $0.86585 = 8.6585 \times 10^{-1}$, so:
$8.6585 \times 10^{-1} \times 10^5 = 8.6585 \times 10^4$

✔ Answer: $8.6585 \times 10^4$

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8) $\frac{5.4 \times 10^{-4}}{3.4 \times 10^1}$

Coefficients:
$\frac{5.4}{3.4} \approx 1.5882$

Exponents:
$-4 - 1 = -5$

So: $1.5882 \times 10^{-5}$

✔ Answer: $1.5882 \times 10^{-5}$

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9) $\frac{4 \times 10^4}{3.63 \times 10^{-4}}$

Coefficients:
$\frac{4}{3.63} \approx 1.1019$

Exponents:
$4 - (-4) = 8$

So: $1.1019 \times 10^8$

✔ Answer: $1.1019 \times 10^8$

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10) $\frac{9 \times 10^{-5}}{9.24 \times 10^{-6}}$

Coefficients:
$\frac{9}{9.24} \approx 0.9739$

Exponents:
$-5 - (-6) = 1$

So: $0.9739 \times 10^1 = 9.739$

But $9.739 = 9.739 \times 10^0$, so:

✔ Answer: $9.739 \times 10^0$ or simply $9.739$

(But since it's scientific notation, we can write as $9.739 \times 10^0$)

✔ Answer: $9.739 \times 10^0$

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11) $\frac{8.42 \times 10^7}{5 \times 10^2}$

Coefficients:
$\frac{8.42}{5} = 1.684$

Exponents:
$7 - 2 = 5$

So: $1.684 \times 10^5$

✔ Answer: $1.684 \times 10^5$

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12) $\frac{8.9 \times 10^6}{8.4 \times 10^6}$

Coefficients:
$\frac{8.9}{8.4} \approx 1.0595$

Exponents:
$6 - 6 = 0$

So: $1.0595 \times 10^0$

✔ Answer: $1.0595 \times 10^0$

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13) $(8.9 \times 10^{-3})^4$

Apply exponent to both parts:
$(8.9)^4 \times (10^{-3})^4$

First:
$8.9^2 = 79.21$
$8.9^4 = (79.21)^2 = 6274.3241$

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

So: $6274.3241 \times 10^{-12}$ → adjust

$6274.3241 = 6.2743241 \times 10^3$

So: $6.2743241 \times 10^3 \times 10^{-12} = 6.2743241 \times 10^{-9}$

✔ Answer: $6.274 \times 10^{-9}$ (rounded to 4 sig figs)

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14) $(4 \times 10^{-4})^{-6}$

Apply exponent:
$4^{-6} \times (10^{-4})^{-6}$

First:
$4^{-6} = \frac{1}{4^6} = \frac{1}{4096} \approx 0.00024414$

Second:
$(10^{-4})^{-6} = 10^{24}$

So: $0.00024414 \times 10^{24} = 2.4414 \times 10^{-4} \times 10^{24} = 2.4414 \times 10^{20}$

✔ Answer: $2.4414 \times 10^{20}$

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Final Answers:

1) $9.984 \times 10^{-12}$
2) $1.62 \times 10^{-2}$
3) $9.28 \times 10^{-9}$
4) $3.4104 \times 10^{10}$
5) $4.757 \times 10^{-8}$
6) $5.946 \times 10^9$
7) $8.6585 \times 10^4$
8) $1.5882 \times 10^{-5}$
9) $1.1019 \times 10^8$
10) $9.739 \times 10^0$
11) $1.684 \times 10^5$
12) $1.0595 \times 10^0$
13) $6.274 \times 10^{-9}$
14) $2.4414 \times 10^{20}$

Let me know if you'd like these rounded to specific significant figures!