Middle School Math Quiz: Estimate With Powers Of 10
20 questions · exam conditions
0:00
Estimate With Powers Of 10Question 1 of 20

The average distance from Earth to the Moon is about 4×1054\times 10^5 km, and the average distance from Earth to the Sun is about 1.5×1081.5\times 10^8 km. About how many times farther is the Sun than the Moon?

About 3.75×1013.75\times 10^1 times
About 3.75×10133.75\times 10^{13} times
About 3.75×1023.75\times 10^2 times
About 3.75×1033.75\times 10^3 times
← Back to quizzes

Middle School Math Quiz

Middle School Math Quiz: Estimate With Powers Of 10

Practice Estimate With Powers Of 10 in Middle School Math with focused quiz questions that help you check what you know, review explanations, and build confidence with test-style prompts.

What this quiz covers

This quiz focuses on Estimate With Powers Of 10, giving you a quick way to practice the rules, question types, and explanations that matter most for Middle School Math.

How to use this quiz

Try each quiz question before looking at the correct answer. Use the explanations to review missed ideas, then come back to similar questions until the pattern feels familiar.

All questions

Question 1

The average distance from Earth to the Moon is about 4×1054\times 10^5 km, and the average distance from Earth to the Sun is about 1.5×1081.5\times 10^8 km. About how many times farther is the Sun than the Moon?

  1. About 3.75×1013.75\times 10^1 times
  2. About 3.75×10133.75\times 10^{13} times
  3. About 3.75×1023.75\times 10^2 times (correct answer)
  4. About 3.75×1033.75\times 10^3 times
Explanation: This question tests estimating large distances as single digit × 10ⁿ and comparing using division to find 'how many times' farther. Scientific notation a×10ⁿ estimates quantities (Moon 4×1054×10^5 km, Sun 1.5×1081.5×10^8 km). To compare magnitudes: divide (1.5×10⁸)/(4×1054×10^5), separate: (1.5/4)×(108/10510⁸/10^5), calculate: 0.375×10³=3.75×10², about 375 times. For example, 1.5 divided by 4 is 0.375, and 10^8 divided by 10^5 is 10^3, so 0.375×1000=375. This is correct because it properly handles the coefficient less than 1 by adjusting the exponent. A common error is forgetting to adjust the scientific notation or adding exponents instead of subtracting. The process is: (1) express in a×10ⁿ form, (2) divide quantities, (3) divide coefficients (1.5÷4=0.375), subtract exponents (8-5=3), (4) multiply (0.375×103=3.75×1020.375×10^3=3.75×10^2), (5) interpret as about 375 times farther.

Question 2

The population of the United States is about 3×1083\times 10^8 people, and the world population is about 7×1097\times 10^9 people. About how many times larger is the world population than the U.S. population?

  1. About 1010 times
  2. About 2020 times (correct answer)
  3. About 101710^{17} times
  4. About 22 times
Explanation: This question tests estimating very large quantities as single digit × 10ⁿ and comparing using division to find 'how many times' larger. Scientific notation a×10ⁿ estimates quantities (US about 331 million ≈ 3×10⁸, rounding 331 to 3, million=10⁶ but 3×100 million=3×10⁸). To compare magnitudes: divide (7×10⁹)/(3×10⁸), separate: (7/3)×(10⁹/10⁸), calculate: ≈2.33×10¹ which is about 23 times larger, close to 20. For example, 7 divided by 3 is approximately 2.3, and 10^9 divided by 10^8 is 10^1, so 2.3×10=23. This is correct because it accounts for both coefficients and exponents properly. A common error is ignoring coefficients and just subtracting exponents, getting 10 times, or adding exponents. The process is: (1) express in a×10ⁿ form (already given), (2) divide quantities, (3) divide coefficients (7÷3≈2.3), divide powers of 10 (subtract exponents: 9-8=1), (4) multiply results (2.3×10), (5) interpret as about 20 times larger.

Question 3

Order these distances from smallest to largest: 3×10−43\times 10^{-4} m, 6×10−26\times 10^{-2} m, 2×10−32\times 10^{-3} m, 5×10−55\times 10^{-5} m.​

  1. 5×10−55\times 10^{-5}, 3×10−43\times 10^{-4}, 2×10−32\times 10^{-3}, 6×10−26\times 10^{-2} (correct answer)
  2. 6×10−26\times 10^{-2}, 2×10−32\times 10^{-3}, 3×10−43\times 10^{-4}, 5×10−55\times 10^{-5}
  3. 3×10−43\times 10^{-4}, 5×10−55\times 10^{-5}, 2×10−32\times 10^{-3}, 6×10−26\times 10^{-2}
  4. 5×10−55\times 10^{-5}, 2×10−32\times 10^{-3}, 3×10−43\times 10^{-4}, 6×10−26\times 10^{-2}
Explanation: This question tests ordering small quantities expressed as a×10ⁿ from smallest to largest without division. Scientific notation helps compare by looking at exponents first, then coefficients (5×10−55×10^{-5}, 3×10−43×10^{-4}, 2×10−32×10^{-3}, 6×10−26×10^{-2}). Start with the most negative exponent (-5 is smallest), then -4, -3, -2. For example, 10^{-5} is smaller than 10^{-4}, and coefficients adjust the order within same exponents, but here all different. This is correct because smaller exponents (more negative) indicate smaller values. A common error is confusing negative exponents and thinking larger negative means larger value. The process is: (1) list the numbers, (2) compare exponents from most negative to least, (3) if exponents tie, compare coefficients, (4) arrange accordingly, (5) verify sequence: 5×10^{-5}, 3×10^{-4}, 2×10^{-3}, 6×10^{-2}.

Question 4

A school fundraiser collected about 5×1035\times10^3 cans of food. The city food bank collected about 2×1062\times10^6 cans in the same month. About how many times more cans did the food bank collect than the school fundraiser?

  1. About 2.5×10−32.5\times10^{-3} times more
  2. About 4×1034\times10^3 times more
  3. About 4×1094\times10^9 times more
  4. About 4×1024\times10^2 times more (correct answer)
Explanation: This question tests estimating quantities as a single digit times a power of 10 and comparing using division for how many times more one collected. Scientific notation a×10ⁿ estimates, with food bank at 2×10^6 and school at 5×10^3. To compare, divide bank by school: (2×1062×10^6)/(5×1035×10^3) = (2/5) × (106/10310^6/10^3) = 0.4 × 10^3 = 4×10^2 or 400 times. This is correct because 4×10^2 matches option A. A common error is multiplying instead of dividing, or adding exponents. The process is express in a×10ⁿ (given), divide, divide coefficients (2÷5=0.4), subtract exponents (6-3=3), adjust to 4×10^2, interpret as about 400 times more. Pitfalls include inverting the ratio or ignoring coefficient adjustment.

Question 5

A website estimates there are about 6×10236\times10^{23} molecules in a sample of water and about 3×10223\times10^{22} molecules in a smaller sample. About how many times more molecules are in the larger sample than in the smaller sample?

  1. About 2020 times (correct answer)
  2. About 104510^{45} times
  3. About 22 times
  4. About 1010 times
Explanation: This question tests estimating extremely large quantities as single digit × 10ⁿ and comparing using division to find 'how many times' more. Scientific notation a×10ⁿ estimates molecules, with 6×10^{23} and 3×10^{22} in form. To compare, divide (6×10236×10^{23})/(3×10223×10^{22}), separate into (6/3)×(1023/102210^{23}/10^{22}), and calculate 2×10^1 = 20 times. For example, Avogadro-scale numbers like 600 sextillion over 30 sextillion is 20. This is correct as it handles high exponents via subtraction. Common errors include comparing only coefficients or mistaking exponent addition for multiplication. The process is: (1) express in a×10ⁿ, (2) divide, (3) divide coefficients (6÷3=2), subtract exponents (23-22=1), (4) multiply (2×10), (5) interpret as about 20 times more.

Question 6

A school science poster says the U.S. population is about 3×1083\times10^8 people and the world population is about 7×1097\times10^9 people. About how many times larger is the world population than the U.S. population?

  1. About 2 times larger
  2. About 10 times larger
  3. About 20 times larger (correct answer)
  4. About 200 times larger
Explanation: To compare the two populations, divide the world population by the U.S. population: (7×109)/(3×108)=(7/3)×109−8≈2.3×101≈23(7\times10^9)/(3\times10^8) = (7/3)\times10^{9-8} \approx 2.3\times10^1 \approx 23. So the world population is roughly 20 times larger, matching choice C. Choice A is wrong because it only compares the leading digits, 7 versus 3, without accounting for the difference in powers of ten. Choice B is wrong because it only compares the powers of ten, 10910^9 versus 10810^8, without accounting for the coefficients. Choice D overestimates the size of the difference between the powers of ten.

Question 7

Which number is greatest?

  1. 9×1069\times10^6
  2. 7×1077\times10^7
  3. 1×1081\times10^8 (correct answer)
  4. 6×1076\times10^7
Explanation: This question tests comparing quantities expressed as a single digit times a power of 10 to determine which is greatest by evaluating magnitudes. Scientific notation a×10na \times 10^n helps compare by looking at exponents first, then coefficients if exponents are equal. Comparing: 6×107=60,000,0006 \times 10^7 = 60,000,000; 9×106=9,000,0009 \times 10^6 = 9,000,000; 1×108=100,000,0001 \times 10^8 = 100,000,000; 7×107=70,000,0007 \times 10^7 = 70,000,000, so 1×1081 \times 10^8 is largest. This is correct because 10810^8 is a higher power than 10710^7 or 10610^6, and coefficient 1 times 10810^8 is still bigger than 9×1069 \times 10^6 or others, matching C. A common error is thinking higher coefficient always wins without considering the exponent. The process is to convert to standard form or compare exponents (highest is 88), then coefficients if tied, identifying 1×1081 \times 10^8 as greatest. Pitfalls include misreading exponents or assuming all are equal powers.

Question 8

In a space unit, a student reads that the Earth–Moon distance is about 4×1054\times10^5 km and the Earth–Sun distance is about 1.5×1081.5\times10^8 km. About how many times farther is the Sun than the Moon (from Earth)?

  1. About 4,0004{,}000 times farther
  2. About 4040 times farther
  3. About 0.0040.004 times as far
  4. About 400400 times farther (correct answer)
Explanation: This question tests estimating large distances as a single digit times a power of 10 and comparing them using division to find out how many times farther one is than the other. Scientific notation a×10ⁿ estimates quantities, such as rounding 1.5 to 2 for a rough estimate, but here we keep the given coefficients for accuracy in comparison. To compare, divide Earth-Sun by Earth-Moon: (1.5×10⁸)/(4×10⁵) = (1.5/4) × (10⁸/10⁵) = 0.375 × 10³ = 3.75 × 10², which is 375, or about 400 times farther. This is correct because 375 is closest to 400 among the choices, matching option B. A common error is subtracting exponents incorrectly or forgetting to adjust the coefficient after division. The process is to express in a×10ⁿ form (already given), divide the quantities, divide coefficients (1.5÷4=0.375), subtract exponents (8-5=3), multiply results (0.375×10³=375), and interpret as about 400 times. Pitfalls include confusing multiplication with division for ratios, ignoring exponents and just dividing coefficients, or wrong exponent subtraction.

Question 9

A typical cell is about 1×10−51\times 10^{-5} meters across, and a typical atom is about 1×10−101\times 10^{-10} meters across. About how many times larger is the cell than the atom?

  1. 10010^0 times
  2. 101510^{15} times
  3. 10510^5 times (correct answer)
  4. 10−510^{-5} times
Explanation: This question tests estimating very small quantities as single digit × 10ⁿ and comparing using division to find 'how many times' larger. Scientific notation a×10ⁿ estimates quantities (cell 1×10−51×10^{-5} m, atom 1×10−101×10^{-10} m). To compare magnitudes: divide (1×10−51×10^{-5})/(1×10−101×10^{-10}), separate: (1/1)×(10−5/10−1010^{-5}/10^{-10}), calculate: 1×10^{5}, exactly 10^5 times. For example, coefficients are both 1, and dividing powers gives 10^{-5 - (-10)}=10^5. This is correct because negative exponents are handled by adding the absolute values in subtraction. A common error is subtracting exponents without considering the negative signs, getting 10^{-15}. The process is: (1) express in a×10ⁿ form, (2) divide quantities, (3) divide coefficients (1÷1=1), subtract exponents (-5 - (-10)=5), (4) multiply (1×1051×10^5), (5) interpret as 100,000 times larger.

Question 10

The average distance from Earth to the Moon is about 4×1054\times 10^5 km, and the average distance from Earth to the Sun is about 1.5×1081.5\times 10^8 km. About how many times farther is the Sun than the Moon?​

  1. About 3.75×1023.75\times 10^2 times (correct answer)
  2. About 3.75×10133.75\times 10^{13} times
  3. About 3.75×1033.75\times 10^3 times
  4. About 3.75×1013.75\times 10^1 times
Explanation: This question tests estimating large distances as single digit × 10ⁿ and comparing using division to find 'how many times' farther. Scientific notation a×10ⁿ estimates quantities (Moon 4×1054×10^5 km, Sun 1.5×1081.5×10^8 km). To compare magnitudes: divide (1.5×10⁸)/(4×1054×10^5), separate: (1.5/4)×(108/10510⁸/10^5), calculate: 0.375×10³=3.75×10², about 375 times. For example, 1.5 divided by 4 is 0.375, and 10^8 divided by 10^5 is 10^3, so 0.375×1000=375. This is correct because it properly handles the coefficient less than 1 by adjusting the exponent. A common error is forgetting to adjust the scientific notation or adding exponents instead of subtracting. The process is: (1) express in a×10ⁿ form, (2) divide quantities, (3) divide coefficients (1.5÷4=0.375), subtract exponents (8-5=3), (4) multiply (0.375×103=3.75×1020.375×10^3=3.75×10^2), (5) interpret as about 375 times farther.

Question 11

A bacterium is about 2×10−62\times10^{-6} m long. A human hair is about 8×10−58\times10^{-5} m thick. About how many times thicker is the hair than the bacterium is long?

  1. About 400400 times
  2. About 4040 times (correct answer)
  3. About 4×10−114\times10^{-11} times
  4. About 44 times
Explanation: This question tests estimating small quantities as single digit × 10ⁿ and comparing using division to find 'how many times' thicker. Scientific notation a×10ⁿ estimates sizes, with 8×10^{-5} and 2×10^{-6} in form. To compare, divide (8×10−58×10^{-5})/(2×10−62×10^{-6}), separate into (8/2)×(10−5/10−610^{-5}/10^{-6}), and calculate 4×10^1 = 40 times. For example, 0.00008 m divided by 0.000002 m equals 40. This is correct because subtracting negative exponents gives positive 1. A pitfall is ignoring exponents or adding instead of subtracting, resulting in wrong ratios. The process is: (1) use given notation, (2) divide, (3) divide coefficients (8÷2=4), subtract exponents (-5 - (-6)=1), (4) multiply (4×10), (5) interpret as about 40 times.

Question 12

A video game download is about 8×1098\times10^9 bytes. A music file is about 4×1064\times10^6 bytes. About how many times larger is the game download than the music file?

  1. About 2×10−32\times10^{-3} times larger
  2. About 2×1052\times10^5 times larger
  3. About 2×1032\times10^3 times larger (correct answer)
  4. About 2×10152\times10^{15} times larger
Explanation: This question tests estimating large file sizes as a single digit times a power of 10 and comparing using division to find how many times larger one is. Scientific notation a×10ⁿ estimates quantities, with the game at 8×10^9 and music at 4×10^6, both nearly single-digit. To compare, divide game by music: (8×1098×10^9)/(4×1064×10^6) = (8/4) × (109/10610^9/10^6) = 2 × 10^{3}, exactly 2×10^3 or 2000 times. This is correct because it matches option A directly. A common error is adding exponents instead of subtracting, leading to 10^{15}. The process is to express in a×10ⁿ (given), divide, divide coefficients (8÷4=2), subtract exponents (9-6=3), multiply (2×1032×10^3), and interpret as about 2000 times larger. Pitfalls include comparing only exponents without coefficients, or using negative exponents incorrectly.

Question 13

A video game has about 9×1069\times10^6 active players. Another game has about 3×1073\times10^7 active players. About how many times as many players does the second game have as the first?

  1. About 3030 times as many
  2. About 13\frac{1}{3} as many
  3. About 33 times as many (correct answer)
  4. About 300300 times as many
Explanation: This question tests estimating large quantities as single digit × 10ⁿ and comparing using division to find 'how many times' as many. Scientific notation a×10ⁿ estimates players, with 9×10^6 and 3×10^7 already in form (noting 3×1073×10^7 is like 30×10630×10^6 for comparison). To compare, divide (3×1073×10^7)/(9×1069×10^6), separate into (3/9)×(107/10610^7/10^6), and calculate 0.333×10^1 ≈ 3.33, about 3 times as many. For example, 30 million divided by 9 million is roughly 3.33, supporting about 3 times. This is correct as it uses division for ratio and rounds appropriately. Errors might include adding exponents or reversing the division, leading to fractions like 1/3. The process is: (1) express in a×10ⁿ, (2) divide, (3) divide coefficients (3÷9≈0.33), subtract exponents (7-6=1), (4) multiply (0.33×10=3.3), (5) interpret as about 3 times.

Question 14

A typical cell is about 1×10−51\times10^{-5} meters across, and an atom is about 1×10−101\times10^{-10} meters across. About how many times larger is the cell than the atom?

  1. About 10510^5 times (correct answer)
  2. About 55 times
  3. About 10−510^{-5} times
  4. About 101510^{15} times
Explanation: This question tests estimating very small quantities as single digit × 10ⁿ and comparing using division to find 'how many times' larger. Scientific notation a×10ⁿ estimates tiny sizes, like a cell at 1×10^{-5} m and atom at 1×10^{-10} m, both already in single-digit form. To compare, divide (1×10−51×10^{-5})/(1×10−101×10^{-10}), separate into (1/1)×(10−5/10−1010^{-5}/10^{-10}), and calculate 1×10^{5} = 100,000 times larger. For example, 0.00001 m divided by 0.0000000001 m equals 100,000, or 10^5 times. This is correct because dividing powers of 10 with negative exponents subtracts to positive ( -5 - (-10) = 5 ). Common errors include adding exponents instead of subtracting, or mistaking larger exponents for smaller sizes. The process is: (1) express in a×10ⁿ, (2) divide, (3) divide coefficients (1÷1=1), subtract exponents for powers (-5 - (-10)=5), (4) multiply (1×1051×10^5), (5) interpret as about 10^5 times larger.

Question 15

A runner's step length is about 1×1001\times10^0 meters. The thickness of a sheet of paper is about 1×10−41\times10^{-4} meters. About how many times longer is the step length than the paper is thick?

  1. 10010^0 times longer
  2. 10410^4 times longer (correct answer)
  3. 10−410^{-4} times longer
  4. 10810^8 times longer
Explanation: This question tests estimating quantities across scales as a single digit times a power of 10 and comparing using division for how many times longer. Scientific notation a×10ⁿ estimates, with step at 1×10^0 and paper at 1×10^{-4}. To compare, divide step by paper: (1×1001×10^0)/(1×10−41×10^{-4}) = (1/1) × 10^{(0 - (-4))} = 1 × 10^4. This is correct because it's exactly 10^4 times longer, matching option B. A common error is using negative exponent in the result or adding instead of subtracting. The process involves expressing in a×10ⁿ (given), dividing, dividing coefficients (1÷1=1), subtracting exponents (0 - (-4)=4), multiplying (1×1041×10^4), interpreting as 10^4 times. Pitfalls include confusing length with thickness or mishandling the zero exponent.

Question 16

Order these distances from smallest to largest: 3×10−43\times 10^{-4} m, 6×10−26\times 10^{-2} m, 2×10−32\times 10^{-3} m, 5×10−55\times 10^{-5} m.

  1. 5×10−55\times 10^{-5}, 2×10−32\times 10^{-3}, 3×10−43\times 10^{-4}, 6×10−26\times 10^{-2}
  2. 6×10−26\times 10^{-2}, 2×10−32\times 10^{-3}, 3×10−43\times 10^{-4}, 5×10−55\times 10^{-5}
  3. 5×10−55\times 10^{-5}, 3×10−43\times 10^{-4}, 2×10−32\times 10^{-3}, 6×10−26\times 10^{-2} (correct answer)
  4. 3×10−43\times 10^{-4}, 5×10−55\times 10^{-5}, 2×10−32\times 10^{-3}, 6×10−26\times 10^{-2}
Explanation: This question tests ordering small quantities expressed as a×10ⁿ from smallest to largest without division. Scientific notation helps compare by looking at exponents first, then coefficients (5×10−55×10^{-5}, 3×10−43×10^{-4}, 2×10−32×10^{-3}, 6×10−26×10^{-2}). Start with the most negative exponent (-5 is smallest), then -4, -3, -2. For example, 10^{-5} is smaller than 10^{-4}, and coefficients adjust the order within same exponents, but here all different. This is correct because smaller exponents (more negative) indicate smaller values. A common error is confusing negative exponents and thinking larger negative means larger value. The process is: (1) list the numbers, (2) compare exponents from most negative to least, (3) if exponents tie, compare coefficients, (4) arrange accordingly, (5) verify sequence: 5×10^{-5}, 3×10^{-4}, 2×10^{-3}, 6×10^{-2}.

Question 17

A bacteria cell is about 5×10−65\times 10^{-6} meters long. A human hair is about 7×10−57\times 10^{-5} meters thick. About how many times thicker is the hair than the bacteria cell is long?

  1. About 1.4×1091.4\times 10^9 times
  2. About 22 times
  3. About 1.4×10−11.4\times 10^{-1} times
  4. About 1.4×1011.4\times 10^1 times (correct answer)
Explanation: This question tests estimating small quantities as single digit × 10ⁿ and comparing using division to find 'how many times' thicker. Scientific notation a×10ⁿ estimates quantities (hair 7×10−57×10^{-5} m, bacteria 5×10−65×10^{-6} m). To compare magnitudes: divide (7×10−57×10^{-5})/(5×10−65×10^{-6}), separate: (7/5)×(10−5/10−610^{-5}/10^{-6}), calculate: 1.4×10^1, about 14 times. For example, 7 divided by 5 is 1.4, and 10^{-5} divided by 10^{-6} is 10^1, so 1.4×10=14. This is correct because negative exponents are subtracted correctly to give positive. A common error is adding exponents instead of subtracting, getting 10^{-11}. The process is: (1) express in a×10ⁿ form, (2) divide quantities, (3) divide coefficients (7÷5=1.4), subtract exponents (-5 - (-6)=1), (4) multiply (1.4×1011.4×10^1), (5) interpret as about 14 times thicker.

Question 18

The population of the United States is about 3×1083\times10^8 people, and the world population is about 7×1097\times10^9 people. About how many times larger is the world population than the U.S. population?

  1. Roughly 22 times larger
  2. Roughly 1010 times larger
  3. Roughly 2020 times larger (correct answer)
  4. Roughly 100100 times larger
Explanation: This question tests estimating very large quantities as single digit × 10ⁿ and comparing using division to find 'how many times' larger. Scientific notation a×10ⁿ estimates quantities, like the US population of about 331 million ≈ 3×10⁸ by rounding 331 to 3 and adjusting the exponent to 10⁸ for millions times 100. To compare magnitudes, divide (7×10⁹)/(3×10⁸), separate into (7/3)×(10⁹/10⁸), and calculate 2.333×10¹ ≈ 23 times larger, which rounds to about 20 times. For example, 7 billion divided by 300 million is indeed around 23.3, supporting the estimate of 20 times. This is correct because it accounts for both the coefficients and the exponents properly. A common error is adding exponents instead of subtracting during division, or ignoring coefficients and just comparing exponents. The process is: (1) express in a×10ⁿ form (rounding to 1 significant figure), (2) divide quantities, (3) divide coefficients (7÷3≈2.3), divide powers of 10 (subtract exponents: 9-8=1), (4) multiply results (2.3×10), (5) interpret as about 20 times larger.

Question 19

A science website lists the EarthMoon distance as about 4×1054\times10^5 km and the EarthSun distance as about 1.5×1081.5\times10^8 km. About how many times farther is the Sun than the Moon (from Earth)?

  1. About 400400 times (correct answer)
  2. About 4040 times
  3. About 44 times
  4. About 4×10134\times10^{13} times
Explanation: This question tests estimating large distances as single digit × 10ⁿ and comparing using division to find 'how many times' farther. Scientific notation a×10ⁿ estimates quantities, such as rounding 1.5 to 2 for a rough estimate, but here we use the given 1.5×10⁸ and 4×10⁵. To compare, divide (1.5×10⁸)/(4×10⁵), separate into (1.5/4)×(10⁸/10⁵), and calculate 0.375×10³ = 375, which is about 400 times. For example, 150 million km divided by 400 thousand km is 375, close to 400 for estimation purposes. This is correct as it properly handles the division of both coefficients and exponents. A pitfall is confusing operations, like subtracting exponents incorrectly or comparing only coefficients without adjusting for powers of 10. The process involves: (1) using the given scientific notation, (2) dividing the quantities, (3) dividing coefficients (1.5÷4=0.375), subtracting exponents (8-5=3), (4) multiplying (0.375×1000=375), (5) rounding to about 400 times.

Question 20

A typical cell is about 1×10−51\times 10^{-5} meters across, and a typical atom is about 1×10−101\times 10^{-10} meters across. About how many times larger is the cell than the atom?​

  1. 10510^5 times (correct answer)
  2. 10−510^{-5} times
  3. 101510^{15} times
  4. 10010^0 times
Explanation: This question tests estimating very small quantities as single digit × 10ⁿ and comparing using division to find 'how many times' larger. Scientific notation a×10ⁿ estimates quantities (cell 1×10−51×10^{-5} m, atom 1×10−101×10^{-10} m). To compare magnitudes: divide (1×10−51×10^{-5})/(1×10−101×10^{-10}), separate: (1/1)×(10−5/10−1010^{-5}/10^{-10}), calculate: 1×10^{5}, exactly 10^5 times. For example, coefficients are both 1, and dividing powers gives 10^{-5 - (-10)}=10^5. This is correct because negative exponents are handled by adding the absolute values in subtraction. A common error is subtracting exponents without considering the negative signs, getting 10^{-15}. The process is: (1) express in a×10ⁿ form, (2) divide quantities, (3) divide coefficients (1÷1=1), subtract exponents (-5 - (-10)=5), (4) multiply (1×1051×10^5), (5) interpret as 100,000 times larger.