12 multiple-choice questions, progressively harder.
Compute (6×107)×(5×104)(6 \times 10^{7}) \times (5 \times 10^{4})(6×107)×(5×104) in proper scientific notation.
Solution
Correct answer: A
Multiply the coefficients and add the exponents: 6×5=306 \times 5 = 306×5=30 and 7+4=117 + 4 = 117+4=11, giving 30×101130 \times 10^{11}30×1011. Renormalize, since 30=3×1030 = 3 \times 1030=3×10.
30×1011=3×101×1011=3×101230 \times 10^{11} = 3 \times 10^{1} \times 10^{11} = 3 \times 10^{12}30×1011=3×101×1011=3×1012
The proper form is 3×10123 \times 10^{12}3×1012.
Write 0.000004080.000004080.00000408 in scientific notation.
Correct answer: B
The number is below 111, so the point moves right for a negative exponent. Place it after the first nonzero digit, the 444, and count from 0.000004080.000004080.00000408: that is 666 places right. The inner zero between 444 and 888 stays.
0.00000408=4.08×10−60.00000408 = 4.08 \times 10^{-6}0.00000408=4.08×10−6
Compute 7.2×1098×102\dfrac{7.2 \times 10^{9}}{8 \times 10^{2}}8×1027.2×109 in proper scientific notation.
Correct answer: C
Divide the coefficients and subtract the exponents: 7.28=0.9\frac{7.2}{8} = 0.987.2=0.9 and 9−2=79 - 2 = 79−2=7, giving 0.9×1070.9 \times 10^{7}0.9×107. The coefficient 0.90.90.9 is below 111, so renormalize: 0.9=9×10−10.9 = 9 \times 10^{-1}0.9=9×10−1.
0.9×107=9×10−1×107=9×1060.9 \times 10^{7} = 9 \times 10^{-1} \times 10^{7} = 9 \times 10^{6}0.9×107=9×10−1×107=9×106
The proper form is 9×1069 \times 10^{6}9×106.
Compute (2×10−6)×(8×10−3)(2 \times 10^{-6}) \times (8 \times 10^{-3})(2×10−6)×(8×10−3) in proper scientific notation.
Multiply the coefficients and add the exponents: 2×8=162 \times 8 = 162×8=16 and −6+(−3)=−9-6 + (-3) = -9−6+(−3)=−9, giving 16×10−916 \times 10^{-9}16×10−9. Renormalize, since 16=1.6×1016 = 1.6 \times 1016=1.6×10.
16×10−9=1.6×101×10−9=1.6×10−816 \times 10^{-9} = 1.6 \times 10^{1} \times 10^{-9} = 1.6 \times 10^{-8}16×10−9=1.6×101×10−9=1.6×10−8
The proper form is 1.6×10−81.6 \times 10^{-8}1.6×10−8.
A bacterium is about 2×10−62 \times 10^{-6}2×10−6 m long. Laid end to end, how long a line would 4×1054 \times 10^{5}4×105 of them form?
Correct answer: D
Total length is the length of one times the count: multiply the coefficients and add the exponents.
(2×10−6)×(4×105)=8×10−6+5=8×10−1(2 \times 10^{-6}) \times (4 \times 10^{5}) = 8 \times 10^{-6 + 5} = 8 \times 10^{-1}(2×10−6)×(4×105)=8×10−6+5=8×10−1
The coefficient 888 is in range, so the line is 8×10−1=0.88 \times 10^{-1} = 0.88×10−1=0.8 m long.
Compute 3×1026×10−4\dfrac{3 \times 10^{2}}{6 \times 10^{-4}}6×10−43×102 in proper scientific notation.
Divide the coefficients and subtract the exponents: 36=0.5\frac{3}{6} = 0.563=0.5 and 2−(−4)=62 - (-4) = 62−(−4)=6, giving 0.5×1060.5 \times 10^{6}0.5×106. The coefficient 0.50.50.5 is below 111, so renormalize: 0.5=5×10−10.5 = 5 \times 10^{-1}0.5=5×10−1.
0.5×106=5×10−1×106=5×1050.5 \times 10^{6} = 5 \times 10^{-1} \times 10^{6} = 5 \times 10^{5}0.5×106=5×10−1×106=5×105
Subtracting a negative exponent adds, so 2−(−4)=62 - (-4) = 62−(−4)=6, and the proper form is 5×1055 \times 10^{5}5×105.
A grain of rice weighs about 2×10−22 \times 10^{-2}2×10−2 g. About how many grains are in 6×1046 \times 10^{4}6×104 g of rice?
The number of grains is the total weight divided by the weight of one: divide the coefficients and subtract the exponents.
6×1042×10−2=62×104−(−2)=3×106\frac{6 \times 10^{4}}{2 \times 10^{-2}} = \frac{6}{2} \times 10^{4 - (-2)} = 3 \times 10^{6}2×10−26×104=26×104−(−2)=3×106
Subtracting −2-2−2 adds, giving 4+2=64 + 2 = 64+2=6, so there are about 3×1063 \times 10^{6}3×106 grains.
Compute (4×106)×(2.5×10−2)(4 \times 10^{6}) \times (2.5 \times 10^{-2})(4×106)×(2.5×10−2) in proper scientific notation.
Multiply the coefficients and add the exponents: 4×2.5=104 \times 2.5 = 104×2.5=10 and 6+(−2)=46 + (-2) = 46+(−2)=4, giving 10×10410 \times 10^{4}10×104. Renormalize, since 10=1×1010 = 1 \times 1010=1×10.
10×104=1×101×104=1×10510 \times 10^{4} = 1 \times 10^{1} \times 10^{4} = 1 \times 10^{5}10×104=1×101×104=1×105
The proper form is 1×1051 \times 10^{5}1×105.
Which is the largest?
For these negative exponents, the largest exponent gives the largest number. Compare −3,−4,−3,−2-3, -4, -3, -2−3,−4,−3,−2.
2×10−2>9×10−3>5×10−3>8×10−42 \times 10^{-2} > 9 \times 10^{-3} > 5 \times 10^{-3} > 8 \times 10^{-4}2×10−2>9×10−3>5×10−3>8×10−4
The exponent −2-2−2 is the largest, so 2×10−22 \times 10^{-2}2×10−2 is the largest number.
Write 0.072×1040.072 \times 10^{4}0.072×104 in proper scientific notation.
The coefficient 0.0720.0720.072 is below 111, so this is not proper. Rewrite 0.072=7.2×10−20.072 = 7.2 \times 10^{-2}0.072=7.2×10−2 and combine the powers.
0.072×104=7.2×10−2×104=7.2×1020.072 \times 10^{4} = 7.2 \times 10^{-2} \times 10^{4} = 7.2 \times 10^{2}0.072×104=7.2×10−2×104=7.2×102
Moving the point right two places to fix the coefficient lowers the exponent by 222, from 444 to 222.
Star A is 3×10143 \times 10^{14}3×1014 km away and star B is 6×10126 \times 10^{12}6×1012 km away. About how many times farther is star A?
Divide star A's distance by star B's: divide the coefficients and subtract the exponents.
3×10146×1012=36×1014−12=0.5×102=5×101\frac{3 \times 10^{14}}{6 \times 10^{12}} = \frac{3}{6} \times 10^{14-12} = 0.5 \times 10^{2} = 5 \times 10^{1}6×10123×1014=63×1014−12=0.5×102=5×101
That is 505050, so star A is about 505050 times farther.
Compute 1.5×10−35×102\dfrac{1.5 \times 10^{-3}}{5 \times 10^{2}}5×1021.5×10−3 in proper scientific notation.
Divide the coefficients and subtract the exponents: 1.55=0.3\frac{1.5}{5} = 0.351.5=0.3 and −3−2=−5-3 - 2 = -5−3−2=−5, giving 0.3×10−50.3 \times 10^{-5}0.3×10−5. The coefficient 0.30.30.3 is below 111, so renormalize: 0.3=3×10−10.3 = 3 \times 10^{-1}0.3=3×10−1.
0.3×10−5=3×10−1×10−5=3×10−60.3 \times 10^{-5} = 3 \times 10^{-1} \times 10^{-5} = 3 \times 10^{-6}0.3×10−5=3×10−1×10−5=3×10−6
The proper form is 3×10−63 \times 10^{-6}3×10−6.
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