Significant Figure and Scientific Notation Worksheet with exercises on scientific notation, significant digits, and decimal conversion.

Problem Overview:

The worksheet involves three main tasks:
1. Convert numbers into scientific notation.
2. Determine the number of significant digits in each number.
3. Convert numbers from scientific notation to decimal form.

Let's solve each task step by step.

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Task 1: Convert each of the following into scientific notation.

#### Rules for Scientific Notation:
- A number in scientific notation is written as \( a \times 10^n \), where:
- \( 1 \leq a < 10 \) (the coefficient \( a \) is a single-digit integer followed by a decimal point and any other digits).
- \( n \) is an integer that indicates how many places the decimal point has been moved.

#### Solutions:

a) 3427
- Move the decimal point 3 places to the left: \( 3.427 \times 10^3 \)
- Answer: \( 3.427 \times 10^3 \)

b) 0.00456
- Move the decimal point 3 places to the right: \( 4.56 \times 10^{-3} \)
- Answer: \( 4.56 \times 10^{-3} \)

c) 123,453
- Move the decimal point 5 places to the left: \( 1.23453 \times 10^5 \)
- Answer: \( 1.23453 \times 10^5 \)

d) 172
- Move the decimal point 2 places to the left: \( 1.72 \times 10^2 \)
- Answer: \( 1.72 \times 10^2 \)

e) 0.000984
- Move the decimal point 4 places to the right: \( 9.84 \times 10^{-4} \)
- Answer: \( 9.84 \times 10^{-4} \)

f) 0.502
- Move the decimal point 1 place to the right: \( 5.02 \times 10^{-1} \)
- Answer: \( 5.02 \times 10^{-1} \)

g) 3100.0 × 10²
- Simplify: \( 3100.0 \times 10^2 = 310000 \)
- Move the decimal point 5 places to the left: \( 3.10000 \times 10^5 \)
- Answer: \( 3.10000 \times 10^5 \)

h) 0.0114 × 10⁴
- Simplify: \( 0.0114 \times 10^4 = 114 \)
- Move the decimal point 2 places to the left: \( 1.14 \times 10^2 \)
- Answer: \( 1.14 \times 10^2 \)

i) 107.2
- Move the decimal point 2 places to the left: \( 1.072 \times 10^2 \)
- Answer: \( 1.072 \times 10^2 \)

j) 0.0000455
- Move the decimal point 5 places to the right: \( 4.55 \times 10^{-5} \)
- Answer: \( 4.55 \times 10^{-5} \)

k) 2205.2
- Move the decimal point 3 places to the left: \( 2.2052 \times 10^3 \)
- Answer: \( 2.2052 \times 10^3 \)

l) 30.0 × 10⁻²
- Simplify: \( 30.0 \times 10^{-2} = 0.300 \)
- Move the decimal point 1 place to the right: \( 3.00 \times 10^{-1} \)
- Answer: \( 3.00 \times 10^{-1} \)

m) 0.982 × 10⁻³
- Simplify: \( 0.982 \times 10^{-3} = 0.000982 \)
- Move the decimal point 4 places to the right: \( 9.82 \times 10^{-4} \)
- Answer: \( 9.82 \times 10^{-4} \)

n) 0.0473
- Move the decimal point 2 places to the right: \( 4.73 \times 10^{-2} \)
- Answer: \( 4.73 \times 10^{-2} \)

o) 650,502
- Move the decimal point 5 places to the left: \( 6.50502 \times 10^5 \)
- Answer: \( 6.50502 \times 10^5 \)

p) 3.03 × 10⁻¹
- Already in scientific notation.
- Answer: \( 3.03 \times 10^{-1} \)

q) 20.4 × 10⁵
- Simplify: \( 20.4 \times 10^5 = 2.04 \times 10^6 \)
- Answer: \( 2.04 \times 10^6 \)

r) 1000 × 10⁻³
- Simplify: \( 1000 \times 10^{-3} = 1 \)
- Move the decimal point 0 places: \( 1.0 \times 10^0 \)
- Answer: \( 1.0 \times 10^0 \)

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Task 2: Determine the number of significant digits in each of the following.

#### Rules for Significant Digits:
- Non-zero digits are always significant.
- Zeros between non-zero digits are significant.
- Leading zeros are not significant.
- Trailing zeros are significant if they come after a decimal point or are confirmed by context.

#### Solutions:

a) 3427
- All digits are non-zero.
- Significant digits: 4

b) 0.00456
- Leading zeros are not significant; only the digits 4, 5, and 6 are significant.
- Significant digits: 3

c) 123,453
- All digits are non-zero.
- Significant digits: 6

d) 172
- All digits are non-zero.
- Significant digits: 3

e) 0.000984
- Leading zeros are not significant; only the digits 9, 8, and 4 are significant.
- Significant digits: 3

f) 0.502
- The zero between 5 and 2 is significant.
- Significant digits: 3

g) 3100.0 × 10²
- The trailing zero after the decimal point is significant.
- Significant digits: 5

h) 0.0114 × 10⁴
- Leading zeros are not significant; only the digits 1, 1, and 4 are significant.
- Significant digits: 3

i) 107.2
- All digits are significant.
- Significant digits: 4

j) 0.0000455
- Leading zeros are not significant; only the digits 4, 5, and 5 are significant.
- Significant digits: 3

k) 2205.2
- All digits are significant.
- Significant digits: 5

l) 30.0 × 10⁻²
- The trailing zero after the decimal point is significant.
- Significant digits: 3

m) 0.982 × 10⁻³
- All digits are significant.
- Significant digits: 3

n) 0.0473
- Leading zeros are not significant; only the digits 4, 7, and 3 are significant.
- Significant digits: 3

o) 650,502
- All digits are significant.
- Significant digits: 6

p) 3.03 × 10⁻¹
- All digits are significant.
- Significant digits: 3

q) 20.4 × 10⁵
- All digits are significant.
- Significant digits: 3

r) 1000 × 10⁻³
- Only the digit 1 is significant (trailing zeros are not significant without a decimal point).
- Significant digits: 1

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Task 3: Convert each into decimal form.

#### Rules for Converting from Scientific Notation:
- If \( n > 0 \), move the decimal point \( n \) places to the right.
- If \( n < 0 \), move the decimal point \( |n| \) places to the left.

#### Solutions:

a) 1.56 × 10⁴
- Move the decimal point 4 places to the right: \( 15600 \)
- Answer: \( 15600 \)

b) 0.56 × 10⁻²
- Move the decimal point 2 places to the left: \( 0.0056 \)
- Answer: \( 0.0056 \)

c) 3.69 × 10⁻²
- Move the decimal point 2 places to the left: \( 0.0369 \)
- Answer: \( 0.0369 \)

d) 736.9 × 10⁵
- Move the decimal point 5 places to the right: \( 73690000 \)
- Answer: \( 73690000 \)

e) 0.00259 × 10⁵
- Move the decimal point 5 places to the right: \( 259 \)
- Answer: \( 259 \)

f) 13.69 × 10⁻²
- Move the decimal point 2 places to the left: \( 0.1369 \)
- Answer: \( 0.1369 \)

g) 6.9 × 10⁴
- Move the decimal point 4 places to the right: \( 69000 \)
- Answer: \( 69000 \)

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Final Answers:

1. Scientific Notation:
- a) \( 3.427 \times 10^3 \)
- b) \( 4.56 \times 10^{-3} \)
- c) \( 1.23453 \times 10^5 \)
- d) \( 1.72 \times 10^2 \)
- e) \( 9.84 \times 10^{-4} \)
- f) \( 5.02 \times 10^{-1} \)
- g) \( 3.10000 \times 10^5 \)
- h) \( 1.14 \times 10^2 \)
- i) \( 1.072 \times 10^2 \)
- j) \( 4.55 \times 10^{-5} \)
- k) \( 2.2052 \times 10^3 \)
- l) \( 3.00 \times 10^{-1} \)
- m) \( 9.82 \times 10^{-4} \)
- n) \( 4.73 \times 10^{-2} \)
- o) \( 6.50502 \times 10^5 \)
- p) \( 3.03 \times 10^{-1} \)
- q) \( 2.04 \times 10^6 \)
- r) \( 1.0 \times 10^0 \)

2. Number of Significant Digits:
- a) 4
- b) 3
- c) 6
- d) 3
- e) 3
- f) 3
- g) 5
- h) 3
- i) 4
- j) 3
- k) 5
- l) 3
- m) 3
- n) 3
- o) 6
- p) 3
- q) 3
- r) 1

3. Decimal Form:
- a) \( 15600 \)
- b) \( 0.0056 \)
- c) \( 0.0369 \)
- d) \( 73690000 \)
- e) \( 259 \)
- f) \( 0.1369 \)
- g) \( 69000 \)

Boxed Final Answer:
\[
\boxed{
\begin{array}{ll}
\text{1. Scientific Notation:} & \text{Answers above} \\
\text{2. Significant Digits:} & \text{Answers above} \\
\text{3. Decimal Form:} & \text{Answers above}
\end{array}
}
\]