Multiplying in Scientific Notation

Show your work for each problem.

1.
Which is the product $(2 \times 10^3)(3 \times 10^4)$ ?
1. a) $6 \times 10^{7}$
2. b) $6 \times 10^{12}$
3. c) $5 \times 10^{7}$
4. d) $6 \times 10^{1}$
Answer: $6 \times 10^{7}$
$2 \times 3 = 6$ and $3 + 4 = 7$ , so the product is $6 \times 10^7$ . Multiplying the exponents would give $10^{12}$ , which is wrong.
2.
$10^4 \times 10^6 = 10^{\square}$ . What number goes in the box?
Answer: 10
Add the exponents: $4 + 6 = 10$ , so $10^4 \times 10^6 = 10^{10}$ .
3.
$(3 \times 10^5)(2 \times 10^2) = \square \times 10^7$ . What number goes in the box?
Answer: 6
Multiply the coefficients: $3 \times 2 = 6$ . The exponents give $5 + 2 = 7$ , so the product is $6 \times 10^7$ .
4.
When you multiply $10^m \times 10^n$ , what do you do with the exponents?
1. a)Subtract them
2. b)Keep the larger one
3. c)Add them
4. d)Multiply them
Answer: Add them
Multiplying powers with the same base adds the exponents: $10^m \times 10^n = 10^{m+n}$ . For example $10^2 \times 10^3 = (10 \cdot 10)(10 \cdot 10 \cdot 10) = 10^5$ .
5.
Calculate $(4 \times 10^{2})(2 \times 10^{5})$ .
Answer: $8 × 10^{7}$
1. Multiply the coefficients: $4 \times 2 = ?$ 8
2. Add the exponents: $2 + 5 = ?$ 7
6.
Calculate $(1.5 \times 10^{3})(3 \times 10^{4})$ .
Answer: $4.5 × 10^{7}$
1. Multiply the coefficients: $1.5 \times 3 = ?$ 4.5
2. Add the exponents: $3 + 4 = ?$ 7
7.
$(6 \times 10^2)(4 \times 10^6) = 24 \times 10^8$ . In scientific notation this is $2.4 \times 10^{\square}$ . What number goes in the box?
Answer: 9
$24 = 2.4 \times 10^1$ , so $24 \times 10^8 = 2.4 \times 10^{8+1} = 2.4 \times 10^9$ .
8.
Which is $(6 \times 10^4)(5 \times 10^3)$ written in scientific notation?
1. a) $30 \times 10^{7}$
2. b) $3 \times 10^{8}$
3. c) $3 \times 10^{12}$
4. d) $3 \times 10^{7}$
Answer: $3 \times 10^{8}$
$6 \times 5 = 30$ and $4 + 3 = 7$ give $30 \times 10^7$ . That has the right value but $30$ is not between $1$ and $10$ : $30 \times 10^7 = 3 \times 10^8$ .
9.
Calculate $(5 \times 10^{-4})(6 \times 10^{-3})$ and write the result in scientific notation.
Answer: $3 × 10^{-6}$
1. Multiply the coefficients: $5 \times 6 = ?$ 30
2. Add the exponents: $-4 + (-3) = ?$ -7
3. Normalize $30 \times 10^{-7}$ : what is the new coefficient? 3
4. Normalize $30 \times 10^{-7}$ : what is the final exponent? -6
10.
Calculate $(2.5 \times 10^{-6})(8 \times 10^{9})$ and write the result in scientific notation.
Answer: $2 × 10^{4}$
1. Multiply the coefficients: $2.5 \times 8 = ?$ 20
2. Add the exponents: $-6 + 9 = ?$ 3
3. Normalize $20 \times 10^{3}$ : what is the new coefficient? 2
4. Normalize $20 \times 10^{3}$ : what is the final exponent? 4
11.
$(7 \times 10^{-5})(2 \times 10^{9}) = 1.4 \times 10^{\square}$ . What number goes in the box?
Answer: 5
$7 \times 2 = 14$ and $-5 + 9 = 4$ give $14 \times 10^4$ . Normalize: $14 \times 10^4 = 1.4 \times 10^5$ .
12.
Which is $(3 \times 10^4)^2$ ?
1. a) $6 \times 10^{8}$
2. b) $9 \times 10^{8}$
3. c) $9 \times 10^{6}$
4. d) $3 \times 10^{8}$
Answer: $9 \times 10^{8}$
$(3 \times 10^4)^2 = 3^2 \times 10^{4 \times 2} = 9 \times 10^8$ . Forgetting to square the $3$ gives $3 \times 10^8$ ; adding the exponents gives $10^6$ .
13.
Calculate $(8 \times 10^{5})(9 \times 10^{-2})$ and write the result in scientific notation.
Answer: $7.2 × 10^{4}$
1. Multiply the coefficients: $8 \times 9 = ?$ 72
2. Add the exponents: $5 + (-2) = ?$ 3
3. Normalize $72 \times 10^{3}$ : what is the new coefficient? 7.2
4. Normalize $72 \times 10^{3}$ : what is the final exponent? 4
14.
Calculate $(4 \times 10^{-3})^2$ and write the result in scientific notation.
Answer: $1.6 × 10^{-5}$
1. Raise the coefficient to the power: $4^2 = ?$ 16
2. Multiply the exponents (power of a power): $-3 \times 2 = ?$ -6
3. Normalize $16 \times 10^{-6}$ : what is the new coefficient? 1.6
4. Normalize $16 \times 10^{-6}$ : what is the final exponent? -5
15.
Light travels $3 \times 10^8$ meters per second. Light from the Sun needs about $5 \times 10^2$ seconds to reach the Earth. The distance is $1.5 \times 10^{\square}$ meters. What number goes in the box?
Answer: 11
$(3 \times 10^8)(5 \times 10^2) = 15 \times 10^{10} = 1.5 \times 10^{11}$ meters, that is 150 million kilometers.
16.
Calculate $(2 \times 10^{3})(5 \times 10^{4})(3 \times 10^{-2})$ and write the result in scientific notation.
Answer: $3 × 10^{6}$
1. Multiply the coefficients: $2 \times 5 \times 3 = ?$ 30
2. Add the exponents: $3 + 4 + (-2) = ?$ 5
3. Normalize $30 \times 10^{5}$ : what is the final exponent? 6
17.
A cell is $8 \times 10^{-6}$ m wide. How long is a row of $5 \times 10^4$ cells placed side by side?
1. a) $4 \times 10^{-2}$ m
2. b) $4 \times 10^{-1}$ m
3. c) $4 \times 10^{-24}$ m
4. d) $4 \times 10^{10}$ m
Answer: $4 \times 10^{-1}$ m
$8 \times 5 = 40$ and $-6 + 4 = -2$ give $40 \times 10^{-2}$ . Normalize: $4 \times 10^{-1}$ m, that is $0.4$ m or 40 cm.
18.
Calculate $(6 \times 10^{-4})^2$ and write the result in scientific notation.
Answer: $3.6 × 10^{-7}$
1. Raise the coefficient to the power: $6^2 = ?$ 36
2. Multiply the exponents (power of a power): $-4 \times 2 = ?$ -8
3. Normalize $36 \times 10^{-8}$ : what is the new coefficient? 3.6
4. Normalize $36 \times 10^{-8}$ : what is the final exponent? -7