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Difficulty: EasyExponents, Roots, and Scientific Notation

A certain type of single-celled organism has a length of approximately 2.5×10−62.5 \times 10^{-6} meters. If 4×1034 \times 10^3 of these organisms are placed end-to-end in a straight line, what is the total length of the line, in meters, written in scientific notation?

  1. A
    1.0×10−41.0 \times 10^{-4}
  2. B
    1.0×10−31.0 \times 10^{-3}
  3. 1.0×10−21.0 \times 10^{-2}Answer
  4. D
    1.0×10−81.0 \times 10^{-8}
  5. E
    1.0×10−171.0 \times 10^{-17}

Answer

The total length of the line is 1.0×10−21.0 \times 10^{-2} meters.
To find the total length of the line of organisms, multiply the length of a single organism by the total number of organisms: (2.5×10−6)×(4×103)(2.5 \times 10^{-6}) \times (4 \times 10^3). Grouping the coefficients and powers of ten gives (2.5×4)×(10−6×103)=10×10−3(2.5 \times 4) \times (10^{-6} \times 10^3) = 10 \times 10^{-3}. To write this in standard scientific notation, rewrite 1010 as 1.0×1011.0 \times 10^1, which yields 1.0×101+(−3)=1.0×10−21.0 \times 10^{1 + (-3)} = 1.0 \times 10^{-2}.

Step-by-Step Solution

1
Set up the multiplication of the two values to find the total length.
Total Length =(2.5×10−6)×(4×103)= (2.5 \times 10^{-6}) \times (4 \times 10^3)
To find the combined length of multiple organisms placed end-to-end, multiply the length of one organism by the total number of organisms.
2
Group the coefficients and the powers of 1010 together, then perform the multiplication.
(2.5×4)×(10−6×103)=10×10−6+3=10×10−3(2.5 \times 4) \times (10^{-6} \times 10^3) = 10 \times 10^{-6 + 3} = 10 \times 10^{-3}
Use the associative and commutative properties of multiplication, and add the exponents when multiplying powers with the same base: 10a×10b=10a+b10^a \times 10^b = 10^{a+b}.
3
Convert the result into standard scientific notation a×10na \times 10^n where 1≤∣a∣<101 \le |a| < 10.
10×10−3=1.0×101×10−3=1.0×10−210 \times 10^{-3} = 1.0 \times 10^1 \times 10^{-3} = 1.0 \times 10^{-2}
Since 1010 is not less than 1010, rewrite 1010 as 1.0×1011.0 \times 10^1 and add the exponents (1+(−3)=−21 + (-3) = -2) to express the number in standard scientific notation.

Key Concept

Multiplying numbers in scientific notation and converting to standard scientific notation.
Estimated Time:1m 0s
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Topics

Exponents, Roots, and Scientific Notation

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