12 multiple-choice questions, progressively harder.
Write 63,40063{,}40063,400 in scientific notation.
Solution
Correct answer: A
Put the decimal point after the first nonzero digit, giving the coefficient 6.346.346.34 (drop the trailing zeros). Then count the places the point moves left, from after the last zero in 63,400.63{,}400.63,400. to between the 666 and the 333: that is 444 places.
63,400=6.34×10463{,}400 = 6.34 \times 10^{4}63,400=6.34×104
The coefficient 63.463.463.4 is not allowed, since it is 101010 or more.
Which number is the largest?
Correct answer: D
With each coefficient between 111 and 101010, the number with the largest exponent is largest. Compare the exponents: 4,3,2,54, 3, 2, 54,3,2,5.
3×105>5×104>9×103>8×1023 \times 10^{5} > 5 \times 10^{4} > 9 \times 10^{3} > 8 \times 10^{2}3×105>5×104>9×103>8×102
So 3×1053 \times 10^{5}3×105 is largest, even though its coefficient 333 is small.
Which number is the smallest?
Correct answer: B
For negative exponents, the more negative the exponent, the smaller the number. Compare the exponents: −2,−4,−3,−2-2, -4, -3, -2−2,−4,−3,−2.
2×10−4<5×10−3<7×10−2<9×10−22 \times 10^{-4} < 5 \times 10^{-3} < 7 \times 10^{-2} < 9 \times 10^{-2}2×10−4<5×10−3<7×10−2<9×10−2
The smallest exponent, −4-4−4, gives the smallest number, 2×10−42 \times 10^{-4}2×10−4.
Write 508,000508{,}000508,000 in scientific notation.
Correct answer: C
Put the decimal point after the first nonzero digit for the coefficient 5.085.085.08 (drop the trailing zeros, keep the inner zero). Count the places the point moves left, from after the last zero in 508,000.508{,}000.508,000. to between the 555 and the 000: that is 555 places.
508,000=5.08×105508{,}000 = 5.08 \times 10^{5}508,000=5.08×105
The mass of a dust grain is about 7.5×10−77.5 \times 10^{-7}7.5×10−7 grams. Written out, how many places is the decimal point moved, and in which direction?
A negative exponent shrinks the number, so the decimal point moves left, and the size of the exponent is the number of places.
7.5×10−7=0.000000757.5 \times 10^{-7} = 0.000000757.5×10−7=0.00000075
The point moves 777 places to the left.
Write 0.0005030.0005030.000503 in scientific notation.
The number is below 111, so the point moves right for a negative exponent. Place it after the first nonzero digit, the 555: from 0.0005030.0005030.000503 that is 444 places right. The zero between the 555 and the 333 is significant.
0.000503=5.03×10−40.000503 = 5.03 \times 10^{-4}0.000503=5.03×10−4
Write 7.04×1047.04 \times 10^{4}7.04×104 as an ordinary number.
The exponent 444 moves the decimal point 444 places to the right. After 7.047.047.04 there are two digits past the point, so two zeros fill the rest.
7.04×104=70,4007.04 \times 10^{4} = 70{,}4007.04×104=70,400
The inner zero in 7.047.047.04 stays as a significant digit.
A red blood cell is about 8×10−68 \times 10^{-6}8×10−6 meters wide and a grain of sand is about 5×10−45 \times 10^{-4}5×10−4 meters wide. Which is wider?
Both coefficients are between 111 and 101010, so the larger exponent gives the larger (wider) number. Compare −6-6−6 and −4-4−4.
−4>−6 ⇒ 5×10−4>8×10−6-4 > -6 \;\Rightarrow\; 5 \times 10^{-4} > 8 \times 10^{-6}−4>−6⇒5×10−4>8×10−6
The grain of sand, at 5×10−45 \times 10^{-4}5×10−4 meters, is wider.
Which equals 4.5×1004.5 \times 10^{0}4.5×100?
Any nonzero base to the power 000 is 111, so 100=110^{0} = 1100=1. Multiplying by 111 leaves the coefficient unchanged.
4.5×100=4.5×1=4.54.5 \times 10^{0} = 4.5 \times 1 = 4.54.5×100=4.5×1=4.5
An exponent of 000 means the decimal point does not move at all.
Write 86,000,00086{,}000{,}00086,000,000 in scientific notation.
Put the decimal point after the first nonzero digit for the coefficient 8.68.68.6. Count the places the point moves left, from after the last zero in 86,000,000.86{,}000{,}000.86,000,000. to between the 888 and the 666: that is 777 places.
86,000,000=8.6×10786{,}000{,}000 = 8.6 \times 10^{7}86,000,000=8.6×107
Write 3.9×10−43.9 \times 10^{-4}3.9×10−4 as an ordinary number.
The exponent −4-4−4 moves the decimal point 444 places to the left, padding with zeros.
3.9×10−4=0.000393.9 \times 10^{-4} = 0.000393.9×10−4=0.00039
The 333 lands in the fourth decimal place.
How would you convert 4.2×1074.2 \times 10^{7}4.2×107 to standard form?
A positive exponent makes the number larger, so the decimal point moves to the right, and the exponent gives the number of places.
4.2×107=42,000,0004.2 \times 10^{7} = 42{,}000{,}0004.2×107=42,000,000
The point moves 777 places right, padding with zeros.
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