Operations Scientific Notation - Kuta Software - Free Printable

Let's solve each problem step by step, simplifying expressions involving scientific notation and writing the answers in scientific notation.

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Rules to Remember:

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

4. Negative exponents:
$a^{-n} = \frac{1}{a^n}$

5. Scientific Notation Format:
$a \times 10^n$, where $1 \leq |a| < 10$

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Now, let's go through each problem:

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1) $(1.08 \times 10^{-9})(9.3 \times 10^{-1})$

Multiply coefficients:
$1.08 \times 9.3 = 9.984$

Add exponents:
$-9 + (-1) = -10$

So:
$9.984 \times 10^{-10}$ ✔

> Answer: $9.984 \times 10^{-10}$

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2) $(2 \times 10^{-7})(8.1 \times 10^{-7})$

Coefficients: $2 \times 8.1 = 16.2$

Exponents: $-7 + (-7) = -14$

So: $16.2 \times 10^{-14}$ → but this is not in proper scientific notation.

Convert: $16.2 = 1.62 \times 10^1$, so:

$1.62 \times 10^1 \times 10^{-14} = 1.62 \times 10^{-13}$

> Answer: $1.62 \times 10^{-13}$

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3) $(2.32 \times 10^{-6})(4 \times 10^{-7})$

Coefficients: $2.32 \times 4 = 9.28$

Exponents: $-6 + (-7) = -13$

So: $9.28 \times 10^{-13}$ ✔

> Answer: $9.28 \times 10^{-13}$

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4) $(3.48 \times 10^7)(9.8 \times 10^6)$

Coefficients: $3.48 \times 9.8 = 34.104$

Exponents: $7 + 6 = 13$

So: $34.104 \times 10^{13}$ → not scientific notation

Convert: $34.104 = 3.4104 \times 10^1$

So: $3.4104 \times 10^1 \times 10^{13} = 3.4104 \times 10^{14}$

> Answer: $3.4104 \times 10^{14}$

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5) $(7.1 \times 10^{-5})(6.7 \times 10^{-4})$

Coefficients: $7.1 \times 6.7 = 47.57$

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

So: $47.57 \times 10^{-9}$ → not scientific

$47.57 = 4.757 \times 10^1$, 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) $(6 \times 10^7)(9.91 \times 10^6)$

Coefficients: $6 \times 9.91 = 59.46$

Exponents: $7 + 6 = 13$

So: $59.46 \times 10^{13}$ → not scientific

$59.46 = 5.946 \times 10^1$

So: $5.946 \times 10^1 \times 10^{13} = 5.946 \times 10^{14}$

> Answer: $5.946 \times 10^{14}$

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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 scientific

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

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

Exponents: $-1 - 1 = -2$

So: $1.5882 \times 10^{-2}$ ✔

> Answer: $1.5882 \times 10^{-2}$

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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: $0.9739 \times 10^1 = 9.739 \times 10^0$ → $9.739$

But wait: $0.9739 \times 10^1 = 9.739$, which is $9.739 \times 10^0$

But better to write as $9.739$ → but that's not in scientific notation unless we keep it as $9.739 \times 10^0$

Actually, since $0.9739 \times 10^1 = 9.739$, and $9.739$ is between 1 and 10, so:

> Answer: $9.739 \times 10^0$ or just $9.739$, but standard form is $9.739 \times 10^0$

✔ But usually we don’t write $10^0$ unless required. But since it says "scientific notation", we can write:

> Answer: $9.739 \times 10^0$

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11) $\frac{8.42 \times 10^3}{5 \times 10^2}$

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

Exponents: $3 - 2 = 1$

So: $1.684 \times 10^1$ ✔

> Answer: $1.684 \times 10^1$

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12) $\frac{8.9 \times 10^6}{8.4 \times 10^7}$

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

Exponents: $6 - 7 = -1$

So: $1.0595 \times 10^{-1}$ ✔

> Answer: $1.0595 \times 10^{-1}$

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13) $(8.9 \times 10^5)^2$

Apply power rule:
$(8.9)^2 = 79.21$
$(10^5)^2 = 10^{10}$

So: $79.21 \times 10^{10}$ → not scientific

Convert: $79.21 = 7.921 \times 10^1$

So: $7.921 \times 10^1 \times 10^{10} = 7.921 \times 10^{11}$

> Answer: $7.921 \times 10^{11}$

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14) $(4 \times 10^{-5})^{-4}$

Use rule: $(a \times 10^b)^n = a^n \times 10^{b \cdot n}$

So:
$4^{-4} = \frac{1}{4^4} = \frac{1}{256} \approx 0.00390625$
$(10^{-5})^{-4} = 10^{(-5)(-4)} = 10^{20}$

So: $0.00390625 \times 10^{20}$ → not scientific

Convert $0.00390625 = 3.90625 \times 10^{-3}$

So: $3.90625 \times 10^{-3} \times 10^{20} = 3.90625 \times 10^{17}$

> Answer: $3.90625 \times 10^{17}$

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✔ Final Answers:

1) $9.984 \times 10^{-10}$
2) $1.62 \times 10^{-13}$
3) $9.28 \times 10^{-13}$
4) $3.4104 \times 10^{14}$
5) $4.757 \times 10^{-8}$
6) $5.946 \times 10^{14}$
7) $8.6585 \times 10^4$
8) $1.5882 \times 10^{-2}$
9) $1.1019 \times 10^8$
10) $9.739 \times 10^0$
11) $1.684 \times 10^1$
12) $1.0595 \times 10^{-1}$
13) $7.921 \times 10^{11}$
14) $3.90625 \times 10^{17}$

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Let me know if you'd like these rounded to a certain number of significant figures!