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Difficulty: MediumExponents, Roots, and Scientific Notation

A supercomputer can perform 4.5×10154.5 \times 10^{15} calculations per second. If a complex simulation requires a total of 1.8×10191.8 \times 10^{19} calculations, which of the following is the number of seconds it will take the supercomputer to complete this simulation?

  1. A
    2.5×10−42.5 \times 10^{-4}
  2. B
    4.0×1044.0 \times 10^4
  3. 4.0×1034.0 \times 10^3Answer
  4. D
    2.5×1042.5 \times 10^4
  5. E
    4.0×10334.0 \times 10^{33}

Answer

4.0×1034.0 \times 10^3
To determine the number of seconds required to complete the simulation, divide the total number of calculations needed by the speed of the supercomputer: 1.8×10194.5×1015\frac{1.8 \times 10^{19}}{4.5 \times 10^{15}}. Dividing the coefficients gives 1.84.5=0.4\frac{1.8}{4.5} = 0.4. Subtracting the exponents gives 1019−15=10410^{19 - 15} = 10^4. This yields 0.4×1040.4 \times 10^4. To express this value in proper scientific notation, shift the decimal point one place to the right to obtain 4.04.0. This shift must be balanced by subtracting 1 from the exponent, yielding 4.0×1034.0 \times 10^3.

Step-by-Step Solution

1
Set up the division to find the total time by dividing total calculations by the rate of calculations per second.
Time = 1.8×10194.5×1015\frac{1.8 \times 10^{19}}{4.5 \times 10^{15}} seconds
Dividing the total work by the rate of work per unit of time gives the total time required.
2
Divide the coefficients and subtract the exponents using scientific notation division rules.
Time = 0.4×1019−15=0.4×1040.4 \times 10^{19 - 15} = 0.4 \times 10^4 seconds
When dividing numbers in scientific notation, divide the decimal coefficients and subtract the exponent of the divisor from the exponent of the dividend.
3
Convert the value to standard scientific notation format (a×10na \times 10^n, where 1≤∣a∣<101 \le |a| < 10).
Time = 4.0×1034.0 \times 10^3 seconds
To write 0.4×1040.4 \times 10^4 in standard scientific notation, move the decimal point one position to the right to make the coefficient 4.04.0, which requires decreasing the exponent of 10 by 1.

Key Concept

Division of numbers in scientific notation and normalization to standard form.
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Topics

Exponents, Roots, and Scientific Notation

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