Dividing in Scientific Notation

Show your work for each problem.

1.
Calculate $(6 \times 10^8) \div (2 \times 10^3)$ .
1. a) $3 \times 10^{11}$
2. b) $3 \times 10^{-5}$
3. c) $4 \times 10^5$
4. d) $3 \times 10^5$
Answer: $3 \times 10^5$
$6 \div 2 = 3$ and $10^{8-3} = 10^5$ , so the quotient is $3 \times 10^5$ . Adding the exponents ( $10^{11}$ ) would be multiplication.
2.
$(9 \times 10^7) \div (3 \times 10^2) = 3 \times 10^{\square}$ . What number goes in the box?
Answer: 5
$10^7 \div 10^2 = 10^{7-2} = 10^5$ . The coefficient $9 \div 3 = 3$ is already between 1 and 10, so the answer is $3 \times 10^5$ .
3.
$(8.4 \times 10^6) \div (2 \times 10^2) = \square \times 10^4$ . What number goes in the box?
Answer: 4.2
$8.4 \div 2 = 4.2$ and $10^{6-2} = 10^4$ , so the quotient is $4.2 \times 10^4$ .
4.
Which rule gives the power of ten when you divide $10^m \div 10^n$ ?
1. a) $10^{m-n}$
2. b) $10^{m+n}$
3. c) $10^{m \cdot n}$
4. d) $10^{m \div n}$
Answer: $10^{m-n}$
Dividing cancels $n$ of the $m$ tens, so $10^m \div 10^n = 10^{m-n}$ . Check: $10^5 \div 10^2 = 1000 = 10^3$ .
5.
Calculate $(8 \times 10^9) \div (4 \times 10^5)$ step by step.
Answer: 20000
1. Divide the coefficients: $8 \div 4 = ?$ 2
2. Subtract the exponents: $9 - 5 = ?$ 4
3. Write $2 \times 10^4$ as an ordinary number. 20000
6.
Calculate $(4.5 \times 10^6) \div (1.5 \times 10^2)$ step by step.
Answer: 30000
1. Divide the coefficients: $4.5 \div 1.5 = ?$ 3
2. Subtract the exponents: $6 - 2 = ?$ 4
3. Write $3 \times 10^4$ as an ordinary number. 30000
7.
Which is $(2.8 \times 10^9) \div (7 \times 10^4)$ written correctly in scientific notation?
1. a) $4 \times 10^5$
2. b) $4 \times 10^6$
3. c) $4 \times 10^4$
4. d) $0.4 \times 10^4$
Answer: $4 \times 10^4$
$2.8 \div 7 = 0.4$ and $10^{9-4} = 10^5$ , giving $0.4 \times 10^5$ . Normalize: $0.4 \times 10^5 = 4 \times 10^4$ . ( $0.4 \times 10^4$ has the wrong exponent and is not normalized.)
8.
Calculate $(2 \times 10^3) \div (5 \times 10^7)$ and write it as $4 \times 10^{\square}$ . What is the exponent?
Answer: -5
$(2 \div 5) \times 10^{3-7} = 0.4 \times 10^{-4}$ . Normalize: $0.4 \times 10^{-4} = 4 \times 10^{-5}$ . The exponent is $-5$ .
9.
$(9 \times 10^{-5}) \div (3 \times 10^{-8}) = 3 \times 10^{\square}$ . What number goes in the box?
Answer: 3
$10^{-5} \div 10^{-8} = 10^{-5-(-8)} = 10^{-5+8} = 10^3$ , and $9 \div 3 = 3$ . So the quotient is $3 \times 10^3$ .
10.
Calculate $(1.8 \times 10^7) \div (6 \times 10^3)$ and normalize the result.
Answer: 3000
1. Divide the coefficients: $1.8 \div 6 = ?$ 0.3
2. Subtract the exponents: $7 - 3 = ?$ 4
3. Normalize $0.3 \times 10^4$ to $3 \times 10^{\square}$ . What is the exponent? 3
11.
Calculate $(5 \times 10^{-3}) \div (2 \times 10^4)$ .
Answer: 0.00000025
1. Divide the coefficients: $5 \div 2 = ?$ 2.5
2. Subtract the exponents: $-3 - 4 = ?$ -7
3. Write $2.5 \times 10^{-7}$ as an ordinary decimal number. 0.00000025
12.
A space probe is $9 \times 10^{10}$ m from Earth. Radio signals travel at $3 \times 10^8$ m per second. How many seconds does a signal need to reach the probe?
Answer: 300
1. Divide the coefficients: $9 \div 3 = ?$ 3
2. Subtract the exponents: $10 - 8 = ?$ 2
3. Write $3 \times 10^2$ seconds as an ordinary number. 300
13.
What is $(3 \times 10^4) \div (6 \times 10^{-2})$ in scientific notation?
1. a) $5 \times 10^1$
2. b) $5 \times 10^5$
3. c) $5 \times 10^2$
4. d) $2 \times 10^6$
Answer: $5 \times 10^5$
$3 \div 6 = 0.5$ and $10^{4-(-2)} = 10^6$ , giving $0.5 \times 10^6 = 5 \times 10^5$ . The choice $5 \times 10^1$ comes from forgetting that subtracting $-2$ adds.
14.
Calculate $(2 \times 10^9) \div (7 \times 10^4)$ and write it in scientific notation with the coefficient rounded to 3 significant figures. What is the coefficient?
Answer: 2.86
$2 \div 7 = 0.285714\ldots$ and $10^{9-4} = 10^5$ . Normalize: $2.85714\ldots \times 10^4$ . To 3 significant figures the coefficient is $2.86$ (the fourth digit 7 rounds up), so the result is $\approx 2.86 \times 10^4$ .
15.
A region has $1.04 \times 10^7$ inhabitants on $5.2 \times 10^3$ km². How many people live on one km²?
Answer: 2000
1. Divide the coefficients: $1.04 \div 5.2 = ?$ 0.2
2. Subtract the exponents: $7 - 3 = ?$ 4
3. Normalize $0.2 \times 10^4$ to $2 \times 10^{\square}$ . What is the exponent? 3
4. How many people per km² is $2 \times 10^3$ ? 2000
16.
A country with $9 \times 10^7$ inhabitants spends $4.5 \times 10^{11}$ dollars a year on health care. How much is that per person?
Answer: 5000
1. Divide the coefficients: $4.5 \div 9 = ?$ 0.5
2. Subtract the exponents: $11 - 7 = ?$ 4
3. Normalize $0.5 \times 10^4$ to $5 \times 10^{\square}$ . What is the exponent? 3
4. Write the amount per person, $5 \times 10^3$ dollars, as an ordinary number. 5000
17.
A water drop has a mass of about $5 \times 10^{-5}$ kg, and one water molecule about $3 \times 10^{-26}$ kg. About how many molecules are in the drop?
Answer: $≈ 1.67 × 10^{21}$
1. Divide the coefficients $5 \div 3$ and round to 3 significant figures. 1.67
2. Subtract the exponents: $-5 - (-26) = ?$ 21
3. Written out in full, how many digits does $1.67 \times 10^{21}$ have? 22
18.
What is $(2.4 \times 10^{-3}) \div (8 \times 10^{-9})$ in scientific notation?
1. a) $3 \times 10^{-7}$
2. b) $3 \times 10^6$
3. c) $3 \times 10^5$
4. d) $3 \times 10^{-12}$
Answer: $3 \times 10^5$
$2.4 \div 8 = 0.3$ and $-3 - (-9) = 6$ , giving $0.3 \times 10^6$ . Normalize: $3 \times 10^5$ . The choice $3 \times 10^6$ forgets to lower the exponent when normalizing.