Expressions and Equations: Understanding and Using Scientific Notation (CCSS.8.EE.3)
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Common Core 8th Grade Math › Expressions and Equations: Understanding and Using Scientific Notation (CCSS.8.EE.3)
The average distance from Earth to the Sun is about 149,600,000 kilometers. Which number is the same as this distance written in scientific notation?
$1.496 \times 10^8$
$14.96 \times 10^7$
$1.496 \times 10^7$
$0.1496 \times 10^9$
Explanation
Move the decimal in 149,600,000 to make a number between 1 and 10: 1.496. You moved it 8 places to the left, so multiply by $10^8$. Thus $1.496 \times 10^8$. Choices with a coefficient not between 1 and 10 or with the wrong exponent reflect misplaced decimals.
A certain bacterium is about 0.0000025 meters long. Which number is the same as this length written in scientific notation?
$2.5 \times 10^{-5}$
$2.5 \times 10^{-6}$
$0.25 \times 10^{-6}$
$2.5 \times 10^{6}$
Explanation
To write 0.0000025 in scientific notation, move the decimal 6 places to the right to get 2.5. Moving right makes the exponent negative: $2.5 \times 10^{-6}$. Other choices show common errors: off by one place, wrong sign, or a coefficient less than 1.
The population of the United States is about $3 \times 10^8$, and the population of the world is about $7 \times 10^9$. About how many times as large is the world population as the U.S. population?
$2.3 \times 10^0$
$2.3 \times 10^{-1}$
$2.3 \times 10^{1}$
$23 \times 10^0$
Explanation
Compute $\dfrac{7 \times 10^9}{3 \times 10^8} = \dfrac{7}{3} \times 10^{9-8} \approx 2.3 \times 10^1$. The exponent 1 shows it's about 23 times as large. Distractors use the wrong exponent or an improper coefficient.
A cell has a diameter of 0.000012 meters. Which number is the same as this diameter written in scientific notation?
$12 \times 10^{-6}$
$1.2 \times 10^{-6}$
$1.2 \times 10^{5}$
$1.2 \times 10^{-5}$
Explanation
Move the decimal in 0.000012 five places to the right to get 1.2, so the power of ten is negative five: $1.2 \times 10^{-5}$. Choices with the wrong sign or an improper coefficient (not between 1 and 10) reflect decimal-place errors.
A city has a population of 3,450,000 people. Which is the number written in scientific notation?
$3.45 \times 10^5$
$3.45 \times 10^6$
$34.5 \times 10^5$
$3.45 \times 10^7$
Explanation
Move the decimal in 3,450,000 to make a number between 1 and 10: 3.45. You moved it 6 places to the left, so multiply by $10^6$. Thus $3{,}450{,}000 = 3.45 \times 10^6$. Choices with a coefficient not between 1 and 10 are not in scientific notation, and an exponent of 5 or 7 would place the decimal in the wrong position.
A bacterium is about 0.0000072 meters long. Which is this length written in scientific notation?
$7.2 \times 10^6$
$0.72 \times 10^{-6}$
$7.2 \times 10^{-5}$
$7.2 \times 10^{-6}$
Explanation
To write 0.0000072 in scientific notation, move the decimal 6 places to the right to get 7.2, so use a negative exponent: $7.2 \times 10^{-6}$. A positive exponent would make the number very large, and exponents of $-5$ or using a coefficient not between 1 and 10 place the decimal incorrectly.
Which is the same as 480,000,000 written in scientific notation?
$4.8 \times 10^8$
$4.8 \times 10^7$
$48 \times 10^8$
$0.48 \times 10^9$
Explanation
Place the decimal after the first nonzero digit: 4.8. Count 8 places moved left, so the exponent is 8: $4.8 \times 10^8$. An exponent of 7 makes the number ten times too small; coefficients like 48 or 0.48 are not between 1 and 10 and are not in proper scientific notation.
The average distance from Earth to the Sun is about 150,000,000 kilometers. Which is this distance written in scientific notation?
$1.5 \times 10^7$
$1.5 \times 10^8$
$15 \times 10^7$
$1.5 \times 10^{-8}$
Explanation
Move the decimal in 150,000,000 to get a single nonzero digit: 150,000,000 = 1.5 with the decimal moved 8 places to the left, so multiply by $10^8$. That gives $1.5 \times 10^8$. Choices with 7 or 9 for the exponent miscount the places, a negative exponent would represent a small number, and a coefficient of 15 is not in proper scientific notation.
A certain bacteria cell is about 0.000002 meters long. Which is this length written in scientific notation?
$2 \times 10^{6}$
$0.2 \times 10^{-5}$
$2.0 \times 10^{-5}$
$2 \times 10^{-6}$
Explanation
Starting at 0.000002, move the decimal 6 places to the right to make 2, which means multiply by $10^{-6}$. So the number is $2 \times 10^{-6}$. A positive exponent would make a large number, and exponents of −5 or coefficients not between 1 and 10 reflect a one-place miscount or improper scientific notation.
The population of the United States is about $3 \times 10^8$ and the population of the world is about $7 \times 10^9$. About how many times as large is the world population compared to the U.S. population?
$2 \times 10^1$
$2.3 \times 10^0$
$2.3 \times 10^2$
$2 \times 10^2$
Explanation
Compute the ratio: $\dfrac{7 \times 10^9}{3 \times 10^8} = \dfrac{7}{3} \times 10^{9-8} = 2.33\ldots \times 10^1 \approx 2 \times 10^1$. This means a little over 20 times. Choices showing $10^0$ (about 2.3 times) or $10^2$ (about 230 or 200 times) mis-handle the power of ten.