Middle School Math Quiz: Perform Operations With Scientific Notation

20 questions · exam conditions

0:00

Perform Operations With Scientific NotationQuestion 1 of 20

The distance from Earth to the nearest star (other than the Sun) is approximately $4.24$ light-years. One light-year equals $5.88 \times 10^{12}$ miles. What is the distance to this star in miles, expressed in scientific notation?

2.49 \times 10^{13}

miles

2.49 \times 10^{12}

miles

1.39 \times 10^{12}

miles

1.39 \times 10^{13}

miles

← Back to quizzes

Middle School Math Quiz

Middle School Math Quiz: Perform Operations With Scientific Notation

Practice Perform Operations With Scientific Notation 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 Perform Operations With Scientific Notation, 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

$2.49 \times 10^{13}$ miles (correct answer)
$2.49 \times 10^{12}$ miles
$1.39 \times 10^{12}$ miles
$1.39 \times 10^{13}$ miles

Explanation: Multiply the distance in light-years by miles per light-year:

4.24 \times 5.88 \times 10^{12} = 24.9312 \times 10^{12} = 2.49312 \times 10^{13} \approx 2.49 \times 10^{13}

miles. Choice B fails to adjust the exponent when converting 24.9 to proper scientific notation. Choice C incorrectly divides instead of multiplying. Choice D uses the wrong coefficient calculation.

Question 2

A computer processor can perform $2.8 \times 10^{9}$ operations per second. If a complex calculation requires $1.68 \times 10^{15}$ operations, how many seconds will it take to complete, expressed in scientific notation?

$6.0 \times 10^{5}$ seconds (correct answer)
$6.0 \times 10^{6}$ seconds
$4.7 \times 10^{24}$ seconds
$4.7 \times 10^{6}$ seconds

Explanation: Divide total operations by operations per second:

(1.68 \times 10^{15}) \div (2.8 \times 10^{9}) = \frac{1.68}{2.8} \times 10^{15-9} = 0.6 \times 10^{6} = 6.0 \times 10^{5}

seconds. Choice B fails to convert 0.6 to proper scientific notation. Choice C multiplies instead of dividing. Choice D uses the wrong coefficient from improper division.

Question 3

A student simplifies the expression for a science project: $\frac{\left(4.8\times10^7\right)\left(2.5\times10^{-3}\right)}{6\times10^2}.$ What is the value in scientific notation?

$2.0\times10^2$ (correct answer)
$2.0\times10^{0}$
$2.0\times10^{-2}$
$20\times10^1$

(2\times10^3)\times(4\times10^5)=(2\times4)\times10^{3+5}=8\times10^8

(multiply coefficients, add exponents). Division:

(6\times10^8)\div(2\times10^5)=(6\div2)\times10^{8-5}=3\times10^3

(divide coefficients, subtract exponents). Addition: requires same exponent—

(3\times10^5)+(2\times10^4)=3\times10^5+0.2\times10^5=3.2\times10^5

(convert

2\times10^4

0.2\times10^5

, then add). For this problem, compute numerator

(4.8\times10^7)(2.5\times10^{-3}) = 1.2\times10^5

, then divide by

6\times10^2

to get

2.0\times10^2

. This correct application simplifies the expression in scientific notation as

2.0\times10^2

. A common error is mishandling the exponents in combined operations. Steps: (1) identify operations as multiplication then division, (2) for

\times$/÷: coefficients and exponents separately, (3) handle numerator first, then divide, (4) verify proper form (

1\leq a<10$, adjust if needed), (5) no units here. Common errors: mixing operation rules (adding exponents for addition), forgetting adjustment (coefficients outside 1-10 range), wrong exponent arithmetic.

Question 4

A scientist measures the mass of a virus as $3.2 \times 10^{-18}$ grams and the mass of a bacterium as $9.5 \times 10^{-13}$ grams. How many times greater is the mass of the bacterium than the mass of the virus?

$2.97 \times 10^{5}$ times greater (correct answer)
$2.97 \times 10^{-5}$ times greater
$6.3 \times 10^{5}$ times greater
$6.3 \times 10^{-31}$ times greater

Explanation: To find how many times greater, divide the bacterium mass by the virus mass:

(9.5 \times 10^{-13}) \div (3.2 \times 10^{-18}) = \frac{9.5}{3.2} \times 10^{-13-(-18)} = 2.97 \times 10^{5}

. Choice B uses the wrong exponent sign. Choice C incorrectly adds the coefficients instead of dividing. Choice D multiplies the exponents instead of subtracting.

Question 5

In a science lab, a student counts about $6\times10^6$ bacteria in one sample and $4\times10^5$ bacteria in another sample. About how many bacteria are there in total? Write your answer in scientific notation.

$10\times10^6$
$6.4\times10^6$ (correct answer)
$1.0\times10^{12}$
$6.04\times10^6$

Explanation: This question tests operations with scientific notation: multiply/divide (apply to coefficients and exponents separately), add/subtract (adjust to same exponent first). Multiplication: (2×10³)×(4×10⁵)=(2×4)×10³⁺⁵=8×10⁸ (multiply coefficients, add exponents). Division: (6×10⁸)÷(2×10⁵)=(6÷2)×10⁸⁻⁵=3×10³ (divide coefficients, subtract exponents). Addition: requires same exponent—(3×10⁵)+(2×10⁴)=3×10⁵+0.2×10⁵=3.2×10⁵ (convert 2×10⁴ to 0.2×10⁵, then add). For this problem, add 6×10^6 and 4×10^5 by adjusting the smaller to 0.4×10^6, so (6 + 0.4) × 10^6 = 6.4 × 10^6. This correct application estimates the total bacteria in scientific notation as 6.4×10^6. A common error is not adjusting exponents before adding, like just adding coefficients and exponents separately. Steps: (1) identify operation as addition, (2) for +/-: adjust to same exponent first (shift decimal), (3) add coefficients, (4) verify proper form (1≤a<10, adjust if needed: 12×10⁵→1.2×10⁶), (5) no units here. Common errors: mixing operation rules (adding exponents for addition), forgetting adjustment (coefficients outside 1-10 range), wrong exponent arithmetic.

Question 6

A computer file is $7.2\times10^6$ bytes. A smaller file is $3.5\times10^5$ bytes. What is the difference in size, in scientific notation? $\left(7.2\times10^6\right)-\left(3.5\times10^5\right)$

$7.55\times10^6$
$6.85\times10^1$
$3.7\times10^1$
$6.85\times10^6$ (correct answer)

Explanation: This question tests operations with scientific notation: multiply/divide (apply to coefficients and exponents separately), add/subtract (adjust to same exponent first). Multiplication: (2×10³)×(4×10⁵)=(2×4)×10³⁺⁵=8×10⁸ (multiply coefficients, add exponents). Division: (6×10⁸)÷(2×10⁵)=(6÷2)×10⁸⁻⁵=3×10³ (divide coefficients, subtract exponents). Addition: requires same exponent—(3×10⁵)+(2×10⁴)=3×10⁵+0.2×10⁵=3.2×10⁵ (convert 2×10⁴ to 0.2×10⁵, then add). For this problem, subtract 3.5×10^5 from 7.2×10^6 by adjusting to 0.35×10^6, so (7.2 - 0.35) × 10^6 = 6.85 × 10^6. This correct application gives the difference in scientific notation as 6.85×10^6 bytes. A common error is subtracting exponents instead of adjusting to the same power. Steps: (1) identify operation as subtraction, (2) for +/-: adjust to same exponent first (shift decimal), (3) subtract coefficients, (4) verify proper form (1≤a<10, here it's fine), (5) include units like bytes. Common errors: mixing operation rules (adding exponents for addition), forgetting adjustment (coefficients outside 1-10 range), wrong exponent arithmetic.

Question 7

A school is tracking recycling. On Monday, students collected $3.2\times10^5$ grams of paper, and on Tuesday they collected $4.8\times10^5$ grams. What is the total amount collected, in scientific notation?

$8\times10^0$
$0.8\times10^6$
$8\times10^{10}$
$8.0\times10^5$ (correct answer)

Explanation: This question tests operations with scientific notation: multiply/divide (apply to coefficients and exponents separately), add/subtract (adjust to same exponent first). Multiplication: (2×10³)×(4×10⁵)=(2×4)×10³⁺⁵=8×10⁸ (multiply coefficients, add exponents). Division: (6×10⁸)÷(2×10⁵)=(6÷2)×10⁸⁻⁵=3×10³ (divide coefficients, subtract exponents). Addition: requires same exponent—(3×10⁵)+(2×10⁴)=3×10⁵+0.2×10⁵=3.2×10⁵ (convert 2×10⁴ to 0.2×10⁵, then add). For this problem, add 3.2×10^5 and 4.8×10^5, which already have the same exponent, so (3.2 + 4.8) × 10^5 = 8.0 × 10^5. This correct application gives the total in scientific notation as 8.0×10^5. A common error is treating addition like multiplication by adding exponents instead of adjusting to the same power. Steps: (1) identify operation as addition, (2) for +/-: adjust to same exponent first (not needed here), (3) add coefficients, (4) verify proper form (1≤a<10, here it's fine), (5) include units if given, like grams. Common errors: mixing operation rules (adding exponents for addition), forgetting adjustment (coefficients outside 1-10 range), wrong exponent arithmetic.

Question 8

Evaluate and write your answer in scientific notation: $\left(8.0\times10^6\right)+\left(2.0\times10^5\right)-\left(3.0\times10^6\right).$

$7.0\times10^6$
$5.0\times10^{11}$
$5.2\times10^6$ (correct answer)
$52\times10^5$

(2 \times 10^3) \times (4 \times 10^5) = (2 \times 4) \times 10^{3+5} = 8 \times 10^8

(multiply coefficients, add exponents). Division:

(6 \times 10^8) \div (2 \times 10^5) = (6 \div 2) \times 10^{8-5} = 3 \times 10^3

(divide coefficients, subtract exponents). Addition: requires same exponent—

(3 \times 10^5) + (2 \times 10^4) = 3 \times 10^5 + 0.2 \times 10^5 = 3.2 \times 10^5

(convert

2 \times 10^4

0.2 \times 10^5

, then add); combine for mixed operations. Here, compute

8.0 \times 10^6 + 2.0 \times 10^5 - 3.0 \times 10^6

: convert to

10^6

8.0 + 0.2 - 3.0 = 5.2 \times 10^6

. This correct application adjusts all to the same exponent before operating. A common error is operating on exponents separately without adjustment. Steps: (1) identify mixed add/subtract, (2) for +/-: adjust to same exponent first, (3) verify proper form (5.2 is fine), (4) no units. Common errors: mixing with multiplication rules, or wrong order of operations.

Question 9

A science kit contains $\left(9.6\times10^{-4}\right)$ liters of solution. The instructions say to split it evenly into $\left(3.2\times10^{-4}\right)$ -liter portions. How many portions can be made? (Compute $\left(9.6\times10^{-4}\right)\div\left(3.2\times10^{-4}\right)$ .)

$0.3\times10^1$
$3.0\times10^{-8}$
$12.8\times10^{-8}$
$3.0\times10^0$ (correct answer)

Explanation: This question tests operations with scientific notation: multiply/divide (apply to coefficients and exponents separately), add/subtract (adjust to same exponent first). Multiplication: (2×10³)×(4×10⁵)=(2×4)×10³⁺⁵=8×10⁸ (multiply coefficients, add exponents). Division: (6×10⁸)÷(2×10⁵)=(6÷2)×10⁸⁻⁵=3×10³ (divide coefficients, subtract exponents). Addition: requires same exponent—(3×10⁵)+(2×10⁴)=3×10⁵+0.2×10⁵=3.2×10⁵ (convert 2×10⁴ to 0.2×10⁵, then add). Here, divide 9.6×10^{-4} by 3.2×10^{-4}: coefficients 9.6÷3.2=3.0, exponents 10^{-4 - (-4)}=10^0, so 3.0×10^0. This correct application handles negative exponents by subtracting properly. A common error is subtracting exponents incorrectly, like 10^{-8}. Steps: (1) identify division, (2) for ×/÷: coefficients and exponents separately, (3) verify proper form (3.0 is fine), (4) portions are unitless. Common errors: mishandling negative signs in exponents, or not simplifying to 10^0.

Question 10

A telescope takes $3.0\times10^4$ pictures of stars each night. How many pictures does it take in $2.0\times10^3$ nights? Which of the following expresses the answer in scientific notation?

$6.0\times10^1$
$6.0\times10^{12}$
$5.0\times10^7$
$6.0\times10^7$ (correct answer)

Explanation: This problem asks you to multiply two numbers in scientific notation:

3.0\times10^4

and

2.0\times10^3

. Multiply the coefficients:

3.0 \times 2.0 = 6.0

. Add the exponents:

4 + 3 = 7

. This gives

6.0\times10^7

pictures. Choice B is a common error from multiplying the exponents together (

4\times3=12

) instead of adding them. Choice C incorrectly combines the coefficients using addition instead of multiplication.

Question 11

A student measures two tiny lengths: $0.0045$ meters and $3.2\times10^{-3}$ meters. What is the sum, written in scientific notation?

$7.7 \times 10^{-2}$
$4.82\times10^{-3}$
$7.7\times10^3$
$7.7\times10^{-3}$ (correct answer)

Explanation: To add values written differently, first convert both to the same power of 10. Since

0.0045 = 4.5 \times 10^{-3}

, the sum becomes

4.5\times10^{-3} + 3.2\times10^{-3} = 7.7\times10^{-3}

. The coefficient 7.7 is between 1 and 10, so this is already in proper scientific notation, matching choice D. Choice A uses the right coefficient but the wrong exponent:

7.7\times10^{-2}=0.077

, not

0.0077

. Choice B doesn't match a standard arithmetic error on these two values. Choice C uses a positive exponent, mishandling the negative power of 10.

Question 12

A computer stores two video files. One file is $9.1\times10^8$ bytes and the other is $3.4\times10^7$ bytes. How many more bytes is the larger file than the smaller file? Write your answer in scientific notation.

$9.44\times10^8$
$8.76\times10^{15}$
$8.76\times10^8$ (correct answer)
$5.7\times10^1$

Explanation: This question tests operations with scientific notation: multiply/divide (apply to coefficients and exponents separately), add/subtract (adjust to same exponent first). Multiplication: (2×10³)×(4×10⁵)=(2×4)×10³⁺⁵=8×10⁸ (multiply coefficients, add exponents). Division: (6×10⁸)÷(2×10⁵)=(6÷2)×10⁸⁻⁵=3×10³ (divide coefficients, subtract exponents). Addition: requires same exponent—(3×10⁵)+(2×10⁴)=3×10⁵+0.2×10⁵=3.2×10⁵ (convert 2×10⁴ to 0.2×10⁵, then add); subtraction follows similarly. For this problem, subtract 9.1×10^8 - 3.4×10^7; convert 3.4×10^7 to 0.34×10^8, then 9.1 - 0.34 = 8.76×10^8 bytes. This correct application adjusts for subtraction by aligning exponents. A common error is subtracting exponents instead of adjusting, like 10^{8-7}=10^1, leading to wrong results. Steps: (1) identify operation as subtraction, (2) for ×/÷: not applicable, (3) for +/-: adjust to same exponent first (shift decimal), (4) verify proper form (1≤a<10, 8.76 is fine), (5) include units as bytes. Common errors: mixing operation rules (subtracting exponents for subtraction like division), forgetting adjustment (coefficients outside 1-10 range), wrong exponent arithmetic.

Question 13

A microscope slide has $2.5\times10^3$ bacteria in one drop and $4.0\times10^5$ bacteria in another drop. If the drops are combined, how many bacteria are there in total? Write your answer in scientific notation.

$4.025\times10^5$ (correct answer)
$4.00025\times10^5$
$6.5\times10^8$
$4.25\times10^5$

Explanation: This question tests operations with scientific notation: multiply/divide (apply to coefficients and exponents separately), add/subtract (adjust to same exponent first). Multiplication: (2×10³)×(4×10⁵)=(2×4)×10³⁺⁵=8×10⁸ (multiply coefficients, add exponents). Division: (6×10⁸)÷(2×10⁵)=(6÷2)×10⁸⁻⁵=3×10³ (divide coefficients, subtract exponents). Addition: requires same exponent—(3×10⁵)+(2×10⁴)=3×10⁵+0.2×10⁵=3.2×10⁵ (convert 2×10⁴ to 0.2×10⁵, then add). For this problem, add 2.5×10^3 and 4.0×10^5; convert 2.5×10^3 to 0.025×10^5, then sum: 4.0 + 0.025 = 4.025×10^5 bacteria. This correct application adjusts the smaller number by decreasing the coefficient and increasing the exponent by 2. A common error is not adjusting and just adding coefficients with the larger exponent, like 4.0 + 2.5 = 6.5×10^5, ignoring the difference in magnitude. Steps: (1) identify operation as addition, (2) for ×/÷: not applicable, (3) for +/-: adjust to same exponent first (shift decimal), (4) verify proper form (1≤a<10, 4.025 is fine), (5) no units beyond bacteria. Common errors: mixing operation rules (adding exponents for addition), forgetting adjustment (coefficients outside 1-10 range), wrong exponent arithmetic.

Question 14

A telescope camera records a bright flash with energy $6.0\times10^8$ joules. If the flash lasted $2.0\times10^5$ seconds, what was the average power in watts? (Compute $\left(6.0\times10^8\right)\div\left(2.0\times10^5\right)$ and give the answer in scientific notation.)

$3.0\times10^3$ (correct answer)
$3.0\times10^{13}$
$3.0\times10^{-3}$
$12\times10^3$

Explanation: This question tests operations with scientific notation: multiply/divide (apply to coefficients and exponents separately), add/subtract (adjust to same exponent first). Multiplication: (2×10³)×(4×10⁵)=(2×4)×10³⁺⁵=8×10⁸ (multiply coefficients, add exponents). Division: (6×10⁸)÷(2×10⁵)=(6÷2)×10⁸⁻⁵=3×10³ (divide coefficients, subtract exponents). Addition: requires same exponent—(3×10⁵)+(2×10⁴)=3×10⁵+0.2×10⁵=3.2×10⁵ (convert 2×10⁴ to 0.2×10⁵, then add). Here, divide 6.0×10^8 by 2.0×10^5: coefficients 6.0÷2.0=3.0, exponents 10^{8-5}=10^3, so 3.0×10^3. This correct application subtracts exponents for division. A common error is adding exponents like in multiplication. Steps: (1) identify division, (2) for ×/÷: coefficients and exponents separately, (3) verify proper form (3.0 is fine), (4) result in watts but not specified. Common errors: wrong sign in exponent subtraction, or dividing exponents instead.

Question 15

The population of bacteria in a culture doubles every hour. If the initial population is $7.5 \times 10^{4}$ bacteria, what will be the population after 3 hours?

$6.0 \times 10^{5}$ bacteria (correct answer)
$2.25 \times 10^{5}$ bacteria
$6.0 \times 10^{4}$ bacteria
$1.5 \times 10^{5}$ bacteria

Explanation: After 3 hours, the population is multiplied by

2^{3} = 8

. So:

7.5 \times 10^{4} \times 8 = 60.0 \times 10^{4} = 6.0 \times 10^{5}

bacteria. Choice B incorrectly uses

2 \times 3 = 6

instead of

2^{3} = 8

. Choice C fails to adjust the exponent when converting to proper scientific notation. Choice D only doubles the population once instead of three times.

Question 16

The area of a rectangular computer screen is $8.64 \times 10^{5}$ square millimeters. If the width is $3.6 \times 10^{2}$ millimeters, what is the length?

$2.4 \times 10^{7}$ millimeters
$2.4 \times 10^{3}$ millimeters (correct answer)
$3.11 \times 10^{8}$ millimeters
$5.04 \times 10^{3}$ millimeters

Explanation: When you encounter problems involving area and dimensions of rectangles, remember that Area = Length × Width. Since you know the area and width, you can find the length by dividing: Length = Area ÷ Width. To solve this problem, you need to divide

8.64 \times 10^{5}

3.6 \times 10^{2}

. When dividing numbers in scientific notation, divide the decimal parts separately from the powers of 10. First, divide the decimals:

8.64 ÷ 3.6 = 2.4

. Then divide the powers of 10:

10^{5} ÷ 10^{2} = 10^{5-2} = 10^{3}

. Combining these gives you

2.4 \times 10^{3}

millimeters. Choice A (

2.4 \times 10^{7}

) results from incorrectly adding the exponents instead of subtracting them when dividing. This is a common error when students confuse the rules for multiplication and division of powers. Choice C (

3.11 \times 10^{8}

) appears to come from multiplying the area by the width instead of dividing, which would give you an unrealistic dimension. Choice D (

5.04 \times 10^{3}

) suggests an error in the decimal division, possibly calculating

8.64 + 3.6

instead of

8.64 ÷ 3.6

. The correct answer is B:

2.4 \times 10^{3}

millimeters. Study tip: When dividing numbers in scientific notation, remember to divide the decimal parts and subtract the exponents (not add them). Always check if your answer makes practical sense—the length should be reasonable for a computer screen.

Question 17

A research lab measures the diameter of a red blood cell as $7.2 \times 10^{-6}$ meters. If red blood cells were lined up in a single row, how many would fit across a distance of 0.0036 meters?

$2.0 \times 10^{-9}$ cells
$5.0 \times 10^{8}$ cells
$5.0 \times 10^{2}$ cells (correct answer)
$2.59 \times 10^{4}$ cells

Explanation: When you encounter problems involving very small measurements and scientific notation, you're being tested on division with powers of 10 and your ability to interpret the reasonableness of your answer. To find how many red blood cells fit across 0.0036 meters, you need to divide the total distance by the diameter of one cell. First, convert 0.0036 to scientific notation:

0.0036 = 3.6 \times 10^{-3}

meters. Now divide:

\frac{3.6 \times 10^{-3}}{7.2 \times 10^{-6}} = \frac{3.6}{7.2} \times \frac{10^{-3}}{10^{-6}} = 0.5 \times 10^{3} = 5.0 \times 10^{2}

This equals 500 cells, which makes sense—you're fitting microscopic cells across a distance of about 4 millimeters. Answer A (

2.0 \times 10^{-9}

) represents a fraction of a cell, which is impossible when counting whole cells. This likely results from incorrectly multiplying the given values instead of dividing. Answer B (

5.0 \times 10^{8}

) equals 500 million cells, which is unreasonably large for such a small distance. This error typically occurs from flipping the fraction or mishandling the negative exponents. Answer D (

2.59 \times 10^{4}

) equals about 26,000 cells, which is too large and suggests an error in the decimal division (perhaps using 7.2 ÷ 3.6 instead of 3.6 ÷ 7.2). Study tip: When dividing numbers in scientific notation, divide the coefficients separately from the powers of 10, and always check if your final answer makes practical sense for the real-world situation.

Question 18

A calculator display shows the result of a calculation as 4.7E-8. This same number could also be written as:

$4.7 \times 10^{8}$
$4.7 \times 10^{-8}$ (correct answer)
$47 \times 10^{-9}$
$0.47 \times 10^{-7}$

Explanation: When you see "E" notation on a calculator display, you're looking at scientific notation. The "E" represents "times 10 to the power of," so 4.7E-8 means

4.7 \times 10^{-8}

. Scientific notation expresses numbers as a coefficient (between 1 and 10) multiplied by a power of 10. The exponent tells you how many places to move the decimal point. A negative exponent means you move the decimal point to the left, creating a very small number. So

4.7 \times 10^{-8}

equals 0.000000047. Looking at the answer choices, option B gives us exactly what the calculator is showing:

4.7 \times 10^{-8}

. This is the standard form of scientific notation. Option A shows

4.7 \times 10^{8}

, which would be 470,000,000 – a completely different number because the exponent is positive instead of negative. Option C gives

47 \times 10^{-9}

. While this equals the same decimal value as our original number, it's not proper scientific notation because the coefficient (47) should be between 1 and 10. Option D shows

0.47 \times 10^{-7}

, which also equals the same decimal but again violates scientific notation rules since 0.47 is less than 1. Remember: proper scientific notation always has a coefficient between 1 and 10. When you see calculator E-notation, simply replace the "E" with "× 10^" to convert it to mathematical notation.

Question 19

A science class measured two distances for a model rocket launch: $6.0\times10^6$ cm and $4.0\times10^5$ cm. Find the total distance, written in scientific notation.

$10.0\times10^6$
$6.04\times10^6$
$6.4\times10^6$ (correct answer)
$1.0\times10^{12}$

Explanation: This question tests operations with scientific notation: multiply/divide (apply to coefficients and exponents separately), add/subtract (adjust to same exponent first). Multiplication: (2×10³)×(4×10⁵)=(2×4)×10³⁺⁵=8×10⁸ (multiply coefficients, add exponents). Division: (6×10⁸)÷(2×10⁵)=(6÷2)×10⁸⁻⁵=3×10³ (divide coefficients, subtract exponents). Addition: requires same exponent—(3×10⁵)+(2×10⁴)=3×10⁵+0.2×10⁵=3.2×10⁵ (convert 2×10⁴ to 0.2×10⁵, then add). For this problem, add 6.0×10^6 and 4.0×10^5; convert 4.0×10^5 to 0.4×10^6, then sum: 6.0 + 0.4 = 6.4×10^6 cm. This correct application involves adjusting the smaller exponent by moving the decimal left and increasing the exponent. A common error is adding without adjustment, like 6.0 + 4.0 = 10.0×10^{something}, but that ignores exponents. Steps: (1) identify operation as addition, (2) for ×/÷: not applicable, (3) for +/-: adjust to same exponent first (shift decimal), (4) verify proper form (1≤a<10, 6.4 is fine), (5) include units if given, here cm but not in final answer. Common errors: mixing operation rules (adding exponents for addition), forgetting adjustment (coefficients outside 1-10 range), wrong exponent arithmetic.

Question 20

A microscope slide has $6\times10^8$ cells spread evenly across $2\times10^5$ tiny squares on a grid. About how many cells are in each square? Write your answer in scientific notation.

$3\times10^4$
$3\times10^3$ (correct answer)
$3\times10^{13}$
$12\times10^3$

Explanation: This question tests operations with scientific notation: multiply/divide (apply to coefficients and exponents separately), add/subtract (adjust to same exponent first). Multiplication: (2×10³)×(4×10⁵)=(2×4)×10³⁺⁵=8×10⁸ (multiply coefficients, add exponents). Division: (6×10⁸)÷(2×10⁵)=(6÷2)×10⁸⁻⁵=3×10³ (divide coefficients, subtract exponents). Addition: requires same exponent—(3×10⁵)+(2×10⁴)=3×10⁵+0.2×10⁵=3.2×10⁵ (convert 2×10⁴ to 0.2×10⁵, then add). For this problem, divide 6×10^8 by 2×10^5, so (6 ÷ 2) × 10^(8-5) = 3 × 10^3. This correct application estimates cells per square in scientific notation as 3×10^3. A common error is adding exponents instead of subtracting for division. Steps: (1) identify operation as division, (2) for ×/÷: coefficients and exponents separately, (3) divide coefficients and subtract exponents, (4) verify proper form (1≤a<10, here it's fine), (5) no units specified. Common errors: mixing operation rules (adding exponents for addition), forgetting adjustment (coefficients outside 1-10 range), wrong exponent arithmetic.