Scientific Notation - Pre Algebra

1. The mass of a hydrogen atom is approximately $1.67 \times 10^{-27}$ kilograms. Which of the following best represents the total mass of $6.02 \times 10^{23}$ hydrogen atoms?
   Answer: $1.005 \times 10^{-3}$ kilograms When you encounter a problem involving scientific notation multiplication, you're essentially dealing with two separate calculations: multiplying the decimal numbers and adding the exponents. To find the total mass, multiply the mass of one atom by the number of atoms: $(1.67 \times 10^{-27}) \times (6.02 \times 10^{23})$. First, multiply the decimal parts: $1.67 \times 6.02 = 10.0534$. Then add the exponents: $-27 + 23 = -4$. This gives you $10.0534 \times 10^{-4}$ kilograms. Since scientific notation requires the decimal part to be between 1 and 10, convert $10.0534$ to $1.00534$ and increase the exponent by 1, yielding $1.00534 \times 10^{-3}$ kilograms. Rounding to three significant figures gives $1.005 \times 10^{-3}$, which matches answer choice C. Looking at the wrong answers: Choice A has $10.05 \times 10^{-4}$, which represents the calculation before properly converting to standard scientific notation. Choice B shows $1.005 \times 10^{-4}$, indicating someone forgot to adjust the exponent when converting the decimal part. Choice D appears to result from subtracting exponents instead of adding them ($-27 - 23 = -50$, close to the $-51$ shown). Remember that when multiplying numbers in scientific notation, you multiply the decimal parts and add the exponents. Always convert your final answer to proper scientific notation form where the decimal part is between 1 and 10.
2. A computer processes $3.6 \times 10^{8}$ calculations per second. If it runs continuously for $2.4 \times 10^{4}$ seconds, approximately how many calculations will it complete?
   Answer: $8.64 \times 10^{12}$ calculations Multiply the rate by the time: $(3.6 \times 10^{8}) \times (2.4 \times 10^{4}) = (3.6 \times 2.4) \times 10^{8+4} = 8.64 \times 10^{12}$. Choice A incorrectly adds exponents as $8+4=11$. Choice C incorrectly multiplies coefficients as $3.6 \times 2.4 = 6.0$. Choice D incorrectly divides instead of multiplying the given values.
3. Which of the following numbers, when written in scientific notation, has the smallest coefficient (the number before the power of 10)?
   Answer: $0.0000000312$ written as scientific notation Converting each to scientific notation: A) $0.000729 = 7.29 \times 10^{-4}$, B) $58,400,000 = 5.84 \times 10^{7}$, C) $0.0000000312 = 3.12 \times 10^{-8}$, D) $940,000 = 9.4 \times 10^{5}$. The coefficients are 7.29, 5.84, 3.12, and 9.4 respectively. The smallest coefficient is 3.12. Students might incorrectly think the number with the most decimal places or the smallest absolute value has the smallest coefficient.
4. A nanometer is $10^{-9}$ meters. A micrometer is $10^{-6}$ meters. How many nanometers equal $7.5$ micrometers?
   Answer: $7.5 \times 10^{3}$ nanometers When you encounter unit conversion problems with powers of 10, you're working with scientific notation and need to carefully track the exponents to convert between different scales. To find how many nanometers equal 7.5 micrometers, start by setting up the conversion. Since 1 micrometer = $10^{-6}$ meters and 1 nanometer = $10^{-9}$ meters, you need to find how many nanometers fit into one micrometer. Divide the micrometer size by the nanometer size: $\frac{10^{-6}}{10^{-9}} = 10^{-6-(-9)} = 10^{-6+9} = 10^{3}$. This means 1 micrometer contains $10^{3} = 1,000$ nanometers. Therefore, 7.5 micrometers = $7.5 \times 10^{3}$ nanometers. Choice A ($7.5 \times 10^{-15}$) comes from incorrectly multiplying the exponents: $-6 \times -9 = 54$, but then making sign errors. Choice B ($7.5 \times 10^{-3}$) results from subtracting exponents incorrectly: $-6 - (-9) = 3$, but then using a negative exponent. Choice C ($7.5 \times 10^{15}$) happens when you mistakenly add the absolute values of both exponents: $6 + 9 = 15$. Choice D is correct. Study tip: When converting from larger units to smaller units (micrometers to nanometers), you expect more of the smaller units, so your final number should be larger than what you started with. This eliminates any answer with a negative exponent right away.
5. Which statement correctly compares the numbers $5.7 \times 10^{-6}$ and $9.2 \times 10^{-7}$?
   Answer: $5.7 \times 10^{-6} > 9.2 \times 10^{-7}$ because $-6 > -7$ When comparing numbers in scientific notation with different exponents, the number with the larger exponent is greater (when both are positive or both are negative). Since $-6 > -7$, we have $10^{-6} > 10^{-7}$, so $5.7 \times 10^{-6} > 9.2 \times 10^{-7}$. Converting to decimal form: $5.7 \times 10^{-6} = 0.0000057$ and $9.2 \times 10^{-7} = 0.00000092$. Choice A incorrectly compares only coefficients. Choice C incorrectly states $-6 < -7$. Choice D has wrong reasoning about coefficients.
6. The distance from Earth to the nearest star (other than the Sun) is approximately $4.1 \times 10^{13}$ kilometers. The distance from Earth to the Moon is approximately $380,000$ kilometers. How many times farther is the nearest star than the Moon?
   Answer: Approximately $1.08 \times 10^{8}$ times farther First convert Moon distance to scientific notation: $380,000 = 3.8 \times 10^{5}$. Then divide star distance by Moon distance: $\frac{4.1 \times 10^{13}}{3.8 \times 10^{5}} = \frac{4.1}{3.8} \times 10^{13-5} = 1.08 \times 10^{8}$. Choice B uses incorrect exponent subtraction. Choice C inverts the coefficient division. Choice D multiplies the distances instead of dividing.
7. Scientists estimate there are approximately $8.7 \times 10^{6}$ different species on Earth. If $0.0012\%$ of these species become extinct each year, approximately how many species become extinct annually?
   Answer: $1.044 \times 10^{2}$ species Convert the percentage to decimal form: $0.0012\% = \frac{0.0012}{100} = 0.000012 = 1.2 \times 10^{-5}$. Multiply by total species: $(8.7 \times 10^{6}) \times (1.2 \times 10^{-5}) = (8.7 \times 1.2) \times 10^{6+(-5)} = 10.44 \times 10^{1} = 1.044 \times 10^{2}$. Choice A fails to convert the coefficient to proper scientific notation. Choice C uses $1.2 \times 10^{-3}$ instead of $1.2 \times 10^{-5}$. Choice D uses $1.2 \times 10^{-2}$ for the percentage.
8. The speed of light is approximately $3.0 \times 10^{8}$ meters per second. Light from the Sun takes about $500$ seconds to reach Earth. Based on this information, what is the approximate distance from the Sun to Earth?
   Answer: $1.5 \times 10^{11}$ meters When you see a problem involving speed, time, and distance, remember the fundamental relationship: distance = speed × time. This formula is your key to solving motion problems. To find the distance from the Sun to Earth, multiply the speed of light by the travel time: Distance = $(3.0 \times 10^{8} \text{ m/s}) \times (500 \text{ s})$ When multiplying numbers in scientific notation, multiply the coefficients and add the exponents: $3.0 \times 500 = 1,500 = 1.5 \times 10^{3}$ So: $(3.0 \times 10^{8}) \times (500) = (3.0 \times 10^{8}) \times (5.0 \times 10^{2}) = 1.5 \times 10^{10+2} = 1.5 \times 10^{11}$ meters This confirms answer choice D is correct. Looking at the wrong answers: Choice A gives $6.0 \times 10^{10}$, which suggests multiplying $3.0 \times 500$ incorrectly as $6.0 \times 10^{2}$ instead of $1.5 \times 10^{3}$. Choice B shows $1.5 \times 10^{6}$, likely from dividing instead of multiplying ($\frac{3.0 \times 10^{8}}{500}$). Choice C gives $6.0 \times 10^{5}$, combining both the division error and the coefficient mistake. Study tip: For scientific notation problems, always convert regular numbers to scientific notation first (like $500 = 5.0 \times 10^{2}$), then apply the rules systematically. Double-check whether the problem calls for multiplication or division by thinking about what makes physical sense.
9. A microscope can magnify objects up to $1.25 \times 10^{3}$ times their actual size. If a cell measures $4.8 \times 10^{-5}$ meters in diameter, what will be the apparent diameter when viewed through this microscope?
   Answer: $6.0 \times 10^{-2}$ meters To find the apparent diameter, multiply the actual diameter by the magnification: $(4.8 \times 10^{-5}) \times (1.25 \times 10^{3}) = (4.8 \times 1.25) \times 10^{-5+3} = 6.0 \times 10^{-2}$ meters. Choice B incorrectly subtracts exponents instead of adding. Choice C incorrectly divides instead of multiplying. Choice D has the correct coefficient but wrong approach to the exponent calculation.

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