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Difficulty: MediumExponents, Roots, and Scientific Notation

The diameter of a human red blood cell is approximately 7.0×10−67.0 \times 10^{-6} meters, and the diameter of a typical influenza virus is approximately 1.4×10−71.4 \times 10^{-7} meters. How many times larger is the diameter of the red blood cell than the diameter of the influenza virus?

Answer: 50

Answer

50
To determine how many times larger the diameter of the red blood cell is than the diameter of the influenza virus, divide the larger diameter by the smaller diameter: 7.0×10−61.4×10−7\frac{7.0 \times 10^{-6}}{1.4 \times 10^{-7}}. Dividing the coefficients gives 7.01.4=5\frac{7.0}{1.4} = 5. Dividing the powers of ten using the quotient rule for exponents gives 10−610−7=10−6−(−7)=101\frac{10^{-6}}{10^{-7}} = 10^{-6 - (-7)} = 10^1. Multiplying these results gives 5×101=505 \times 10^1 = 50.

Step-by-Step Solution

1
Set up the ratio of the larger diameter to the smaller diameter
7.0×10−61.4×10−7\frac{7.0 \times 10^{-6}}{1.4 \times 10^{-7}}
To find how many times larger one quantity is than another, divide the larger quantity by the smaller quantity.
2
Divide the decimal coefficients
5.05.0
Dividing 7.07.0 by 1.41.4 simplifies the numerical coefficient.
3
Divide the exponential terms using exponent properties
10110^1
Using the quotient rule for exponents, 10a10b=10a−b\frac{10^a}{10^b} = 10^{a - b}, so 10−6−(−7)=10110^{-6 - (-7)} = 10^1.
4
Combine and simplify the final value
5050
Multiplying the coefficient by the simplified power of ten yields 5.0×10=505.0 \times 10 = 50.

Key Concept

Division of numbers in scientific notation using properties of exponents
Estimated Time:1m 30s
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Topics

Exponents, Roots, and Scientific Notation

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