Standard Form - Scientific Notation worksheet - Free Printable

Problem Explanation:

The task involves two main parts:
1. Writing numbers in standard form (scientific notation): This means expressing numbers as a product of a number between 1 and 10 and a power of 10.
2. Expanding numbers from scientific notation to their standard decimal form: This involves converting numbers written in scientific notation back to their full decimal form.

Let's solve each part step by step.

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Part 1: Write the following in standard form

#### Step-by-Step Solution:
To write a number in standard form, we need to express it as \( a \times 10^n \), where \( 1 \leq a < 10 \) and \( n \) is an integer.

1. a) 374 000 =
- Move the decimal point to the left until there is only one non-zero digit to the left of the decimal point.
- \( 374 000 \rightarrow 3.74 \times 10^5 \)
- Answer: \( 3.74 \times 10^5 \)

2. b) 1200 =
- Move the decimal point to the left until there is only one non-zero digit to the left of the decimal point.
- \( 1200 \rightarrow 1.2 \times 10^3 \)
- Answer: \( 1.2 \times 10^3 \)

3. c) 745.68 =
- Move the decimal point to the left until there is only one non-zero digit to the left of the decimal point.
- \( 745.68 \rightarrow 7.4568 \times 10^2 \)
- Answer: \( 7.4568 \times 10^2 \)

4. d) 29.37 =
- Move the decimal point to the left until there is only one non-zero digit to the left of the decimal point.
- \( 29.37 \rightarrow 2.937 \times 10^1 \)
- Answer: \( 2.937 \times 10^1 \)

5. e) 70 000 =
- Move the decimal point to the left until there is only one non-zero digit to the left of the decimal point.
- \( 70 000 \rightarrow 7.0 \times 10^4 \)
- Answer: \( 7.0 \times 10^4 \)

6. f) 3645.21 =
- Move the decimal point to the left until there is only one non-zero digit to the left of the decimal point.
- \( 3645.21 \rightarrow 3.64521 \times 10^3 \)
- Answer: \( 3.64521 \times 10^3 \)

7. g) 1210.4 =
- Move the decimal point to the left until there is only one non-zero digit to the left of the decimal point.
- \( 1210.4 \rightarrow 1.2104 \times 10^3 \)
- Answer: \( 1.2104 \times 10^3 \)

8. h) 74 006 =
- Move the decimal point to the left until there is only one non-zero digit to the left of the decimal point.
- \( 74 006 \rightarrow 7.4006 \times 10^4 \)
- Answer: \( 7.4006 \times 10^4 \)

9. i) 1 000 =
- Move the decimal point to the left until there is only one non-zero digit to the left of the decimal point.
- \( 1 000 \rightarrow 1.0 \times 10^3 \)
- Answer: \( 1.0 \times 10^3 \)

10. j) 98365 =
- Move the decimal point to the left until there is only one non-zero digit to the left of the decimal point.
- \( 98365 \rightarrow 9.8365 \times 10^4 \)
- Answer: \( 9.8365 \times 10^4 \)

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Part 2: Expand the following

#### Step-by-Step Solution:
To expand a number from scientific notation to its standard decimal form, move the decimal point according to the exponent of 10.

1. a) \( 3.74 \times 10^2 = \)
- The exponent is 2, so move the decimal point 2 places to the right.
- \( 3.74 \rightarrow 374 \)
- Answer: \( 374 \)

2. b) \( 8.2 \times 10^2 = \)
- The exponent is 2, so move the decimal point 2 places to the right.
- \( 8.2 \rightarrow 820 \)
- Answer: \( 820 \)

3. c) \( 3.153 \times 10^3 = \)
- The exponent is 3, so move the decimal point 3 places to the right.
- \( 3.153 \rightarrow 3153 \)
- Answer: \( 3153 \)

4. d) \( 4.789 \times 10^4 = \)
- The exponent is 4, so move the decimal point 4 places to the right.
- \( 4.789 \rightarrow 47890 \)
- Answer: \( 47890 \)

5. e) \( 2.5 \times 10^2 = \)
- The exponent is 2, so move the decimal point 2 places to the right.
- \( 2.5 \rightarrow 250 \)
- Answer: \( 250 \)

6. f) \( 1.79 \times 10^4 = \)
- The exponent is 4, so move the decimal point 4 places to the right.
- \( 1.79 \rightarrow 17900 \)
- Answer: \( 17900 \)

7. g) \( 9.306 \times 10^4 = \)
- The exponent is 4, so move the decimal point 4 places to the right.
- \( 9.306 \rightarrow 93060 \)
- Answer: \( 93060 \)

8. h) \( 5.005 \times 10^5 = \)
- The exponent is 5, so move the decimal point 5 places to the right.
- \( 5.005 \rightarrow 500500 \)
- Answer: \( 500500 \)

9. i) \( 7 \times 10^2 = \)
- The exponent is 2, so move the decimal point 2 places to the right.
- \( 7 \rightarrow 700 \)
- Answer: \( 700 \)

10. j) \( 6 \times 10^4 = \)
- The exponent is 4, so move the decimal point 4 places to the right.
- \( 6 \rightarrow 60000 \)
- Answer: \( 60000 \)

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Final Answers:

Part 1:
1. \( 3.74 \times 10^5 \)
2. \( 1.2 \times 10^3 \)
3. \( 7.4568 \times 10^2 \)
4. \( 2.937 \times 10^1 \)
5. \( 7.0 \times 10^4 \)
6. \( 3.64521 \times 10^3 \)
7. \( 1.2104 \times 10^3 \)
8. \( 7.4006 \times 10^4 \)
9. \( 1.0 \times 10^3 \)
10. \( 9.8365 \times 10^4 \)

Part 2:
1. \( 374 \)
2. \( 820 \)
3. \( 3153 \)
4. \( 47890 \)
5. \( 250 \)
6. \( 17900 \)
7. \( 93060 \)
8. \( 500500 \)
9. \( 700 \)
10. \( 60000 \)

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Boxed Final Answer:

\[
\boxed{
\begin{array}{ll}
\text{Part 1:} & \text{Part 2:} \\
1. \, 3.74 \times 10^5 & 1. \, 374 \\
2. \, 1.2 \times 10^3 & 2. \, 820 \\
3. \, 7.4568 \times 10^2 & 3. \, 3153 \\
4. \, 2.937 \times 10^1 & 4. \, 47890 \\
5. \, 7.0 \times 10^4 & 5. \, 250 \\
6. \, 3.64521 \times 10^3 & 6. \, 17900 \\
7. \, 1.2104 \times 10^3 & 7. \, 93060 \\
8. \, 7.4006 \times 10^4 & 8. \, 500500 \\
9. \, 1.0 \times 10^3 & 9. \, 700 \\
10. \, 9.8365 \times 10^4 & 10. \, 60000 \\
\end{array}
}
\]