Question

The diameter of the average cell in the human body is 1 x 10-5 meters. The diameter of a grain of sand is about 2 x 10-4 meters. The diameter of a human hair is about 0.000025 meters.
Part A (2 points): What is the diameter of a human hair written in scientific notation?

Part B (2 points): How many times longer is the diameter of a grain of sand than the diameter of the average cell in the human body? Show your work.




Part C (2 points): Given that 1 meter = 1 x 109 nanometers, what is the diameter of the average cell in the human body, in nanometers? Show your work.




Part D (2 points): The smallest known bacteria are from the genus Mycoplasma and can measure as small as 300 nanometers in diameter. Given that 1 meter = 1 x 109 nanometers, what is the diameter of the smallest bacteria, in meters? Give your answer in scientific notation and show your work.




Part E (2 points): Very small numbers are often written in exponential form using scientific notation to make them easier to multiply or divide. Explain how to apply the laws of exponents when multiplying and dividing numbers written in scientific notation.

PLEASE EXPLAIN!

Answer 1

PART A:

Scientific notation is given in the form of $A*10^n$
Where $A$ is a value in unit and $n$ is an integer

0.000025 is not a unit number [the number before the decimal point is of zero value]

0.000025 = 2.5 ÷ 100,000

100,000 can be written as 10⁵

0.000025 = 2.5 ÷ 10⁵ ⇒ to write this in form of multiplication form, we change 10⁵ to 10⁻⁵

0.000025 = 2.5 × 10⁻⁵

-------------------------------------------------------------------------------------------------------------

PART B:

The diameter of a grain of sand ÷ the diameter of the average cell in the human body
= [2×10⁻⁴] ÷ [1×10⁻⁵]  
= [2÷1] × [10⁻⁴×10⁻⁵]
= 2×10 ⁽⁻⁴ ⁻ ⁽⁻⁵⁾⁾ = 2 × 10¹
= 20

answer: 20 times bigger
--------------------------------------------------------------------------------------------------------------

PART C

Diameter of average cell in human body = 1 × 10⁻⁵ meter
1 meter = 1 × 10⁹ nanometer

1 × 10⁻⁵ meter = [1 × 10⁻⁵] × [1 × 10⁹] = 1 × 10⁽⁻⁵⁺⁹⁾ = 1 × 10⁴ = 10000 nanometer
------------------------------------------------------------------------------------------------------------

PART D

1 meter = 1 × 10⁹ nanometer
300 nanometer = 300 ÷ [1 × 10⁹] = [3×10²] ÷ [1×10⁹] = [3÷1] × [10²÷10⁷] = 3 × 10⁻⁵

Answer: 3 × 10⁻⁵ meter

-------------------------------------------------------------------------------------------------------------

PART E

Laws of exponents that are used in scientific notation is 

$x^m*x^n=x^{m+n}$
$\frac{x^m}{x^n}=x^{m-n}$