Comparing Numbers in Scientific Notation

Show your work for each problem.

1.
Which number is greater: $8.9 \times 10^6$ or $5.2 \times 10^7$ ?
1. a) $5.2 \times 10^7$
2. b)They are equal
3. c)You cannot tell without writing out the zeros
4. d) $8.9 \times 10^6$
Answer: $5.2 \times 10^7$
Both coefficients are between 1 and 10, so the exponents decide: $7 > 6$ . So $5.2 \times 10^7 = 52{,}000{,}000$ is greater than $8.9 \times 10^6 = 8{,}900{,}000$ , even though $8.9 > 5.2$ .
2.
How many times as large as $10^4$ is $10^7$ ?
Answer: 1000
$\frac{10^7}{10^4} = 10^{7-4} = 10^3 = 1000$ . Each step in the exponent is a factor of 10, and there are 3 steps.
3.
Compare $7.1 \times 10^9$ and $2.3 \times 10^{11}$ .
Answer: $7.1 × 10^{9} < 2.3 × 10^{11}$
1. What is the exponent of $7.1 \times 10^9$ ? 9
2. What is the exponent of $2.3 \times 10^{11}$ ? 11
3. Choose the symbol: $7.1 \times 10^9$ ? $2.3 \times 10^{11}$ <
4.
Which symbol makes this true? $3.4 \times 10^5 \;\square\; 3.9 \times 10^5$
1. a) $>$
2. b) $=$
3. c) $<$
4. d)None of them, they cannot be compared
Answer: $<$
Both have exponent $5$ , so the coefficients decide: $3.4 < 3.9$ . So $3.4 \times 10^5 < 3.9 \times 10^5$ .
5.
Normalize before comparing: $45 \times 10^3 = 4.5 \times 10^n$ . What is $n$ ?
Answer: 4
$45 \times 10^3 = 4.5 \times 10^1 \times 10^3 = 4.5 \times 10^4$ . The coefficient got 10 times smaller, so the exponent goes up by 1: $n = 4$ .
6.
Compare $6.25 \times 10^4$ and $6.3 \times 10^4$ .
Answer: $6.25 × 10^{4} < 6.3 × 10^{4}$
1. Both exponents are the same. Type that exponent. 4
2. Which coefficient is greater, $6.25$ or $6.3$ ? 6.3
3. Choose the symbol: $6.25 \times 10^4$ ? $6.3 \times 10^4$ <
7.
Which number is the smallest?
1. a) $4 \times 10^{-5}$
2. b) $4 \times 10^{-4}$
3. c) $4 \times 10^{-3}$
4. d) $4 \times 10^{-2}$
Answer: $4 \times 10^{-5}$
The coefficients are all $4$ , so the smallest exponent gives the smallest number. Of $-2, -3, -4, -5$ the smallest is $-5$ , so $4 \times 10^{-5} = 0.00004$ is the smallest.
8.
Which is greater, $6 \times 10^{-4}$ or $6 \times 10^{-2}$ ? Type the exponent of the greater number.
Answer: -2
$-2 > -4$ , so $6 \times 10^{-2} = 0.06$ is greater than $6 \times 10^{-4} = 0.0006$ . The exponent of the greater number is $-2$ .
9.
Compare $38 \times 10^5$ and $4.1 \times 10^6$ .
Answer: $38 × 10^{5} < 4.1 × 10^{6}$
1. Normalize: $38 \times 10^5 = 3.8 \times 10^n$ . What is $n$ ? 6
2. Now both exponents are $6$ . Which coefficient is greater, $3.8$ or $4.1$ ? 4.1
3. Choose the symbol: $38 \times 10^5$ ? $4.1 \times 10^6$ <
10.
Which number is equal to $56 \times 10^3$ ?
1. a) $5.6 \times 10^2$
2. b) $5.6 \times 10^4$
3. c) $0.56 \times 10^4$
4. d) $5.6 \times 10^5$
Answer: $5.6 \times 10^4$
$56 \times 10^3 = 56{,}000 = 5.6 \times 10^4$ . $0.56 \times 10^4$ is only $5{,}600$ , and $5.6 \times 10^5$ is $560{,}000$ .
11.
How many times as large as $5 \times 10^5$ is $5 \times 10^9$ ?
Answer: 10000
Same coefficient $5$ , so divide the powers: $10^{9-5} = 10^4 = 10{,}000$ .
12.
Compare $5 \times 10^{-4}$ and $8 \times 10^{-6}$ .
Answer: $5 × 10^{-4} > 8 × 10^{-6}$
1. What is the exponent of $5 \times 10^{-4}$ ? -4
2. What is the exponent of $8 \times 10^{-6}$ ? -6
3. Choose the symbol: $5 \times 10^{-4}$ ? $8 \times 10^{-6}$ >
13.
Compare $0.7 \times 10^{-3}$ and $6.5 \times 10^{-5}$ .
Answer: $0.7 × 10^{-3} > 6.5 × 10^{-5}$
1. Normalize: $0.7 \times 10^{-3} = 7 \times 10^n$ . What is $n$ ? -4
2. Which exponent is greater, $-4$ or $-5$ ? -4
3. Choose the symbol: $0.7 \times 10^{-3}$ ? $6.5 \times 10^{-5}$ >
14.
Which list is ordered from smallest to largest?
1. a) $2.5 \times 10^{-3},\ 31 \times 10^{-4},\ 9 \times 10^{-4},\ 1.2 \times 10^{-2}$
2. b) $9 \times 10^{-4},\ 2.5 \times 10^{-3},\ 31 \times 10^{-4},\ 1.2 \times 10^{-2}$
3. c) $1.2 \times 10^{-2},\ 2.5 \times 10^{-3},\ 9 \times 10^{-4},\ 31 \times 10^{-4}$
4. d) $9 \times 10^{-4},\ 31 \times 10^{-4},\ 2.5 \times 10^{-3},\ 1.2 \times 10^{-2}$
Answer: $9 \times 10^{-4},\ 2.5 \times 10^{-3},\ 31 \times 10^{-4},\ 1.2 \times 10^{-2}$
$31 \times 10^{-4} = 3.1 \times 10^{-3}$ . Exponents: $-4 < -3 = -3 < -2$ , and within $-3$ : $2.5 < 3.1$ . So $9 \times 10^{-4} < 2.5 \times 10^{-3} < 31 \times 10^{-4} < 1.2 \times 10^{-2}$ ( $0.0009 < 0.0025 < 0.0031 < 0.012$ ).
15.
How many times as large as $2 \times 10^{-8}$ is $2 \times 10^{-3}$ ?
Answer: 100000
Same coefficient, so $10^{-3-(-8)} = 10^{5} = 100{,}000$ .
16.
Order from smallest to largest: $3.2 \times 10^5$ , $9.1 \times 10^4$ , $30 \times 10^4$ , $1.05 \times 10^6$ .
Answer: $9.1 × 10^{4} < 30 × 10^{4} < 3.2 × 10^{5} < 1.05 × 10^{6}$
1. Normalize: $30 \times 10^4 = 3 \times 10^n$ . What is $n$ ? 5
2. What is the smallest exponent in the list now? 4
3. Two numbers have exponent $5$ . Which coefficient is smaller, $3$ or $3.2$ ? 3
4. What is the exponent of the largest number? 6
17.
Light travels about $3 \times 10^8$ m/s. Sound in air travels about $3 \times 10^2$ m/s. About how many times as fast is light?
Answer: $1{,}000{,}000$
1. What is the exponent of the speed of light? 8
2. What is the exponent of the speed of sound? 2
3. What is the difference of the exponents? 6
4. The coefficients are both $3$ . How many times as fast is light? Type the number $10^6$ in full. 1000000
18.
A human hair is about $8 \times 10^{-5}$ m wide. A virus is about $1 \times 10^{-7}$ m across. How many times as wide as the virus is the hair?
Answer: $800$
1. Which exponent is greater, $-5$ or $-7$ ? -5
2. What is $-5 - (-7)$ ? 2
3. What is $10^2$ ? 100
4. Include the coefficients $8$ and $1$ : how many times as wide is the hair? 800