Scientific Notation

Indicate which of the following numbers are not correctly written in scientific notation and write them correctly:

1. a) $45,2 \cdot 10^3$
2. b) $0,8 \cdot 10^{-2}$
3. c) $3,1 \cdot 10^5$
4. d) $12 \cdot 10^{-4}$

Write the following decimal numbers in scientific notation:

1. a) $450 000 000$
2. b) $0,000000032$
3. c) $12 500$
4. d) $0,000105$

Express the following numbers given in scientific notation in standard decimal form:

1. a) $3,4 \cdot 10^5$
2. b) $1,89 \cdot 10^{-4}$
3. c) $7 \cdot 10^8$
4. d) $5,02 \cdot 10^{-6}$

Order the following numbers written in scientific notation from least to greatest:

$A = 3,2 \cdot 10^4$

$B = 4,1 \cdot 10^{-3}$

$C = 1,5 \cdot 10^4$

$D = 8,9 \cdot 10^{-4}$

$E = 2 \cdot 10^5$

Determine the order of magnitude (the nearest power of $10$) of the following quantities expressed in scientific notation:

1. a) $4,8 \cdot 10^6$
2. b) $1,1 \cdot 10^{-5}$
3. c) $9,2 \cdot 10^{12}$
4. d) $2,5 \cdot 10^{-8}$

Perform the following multiplications by grouping the numbers and the powers of base $10$. Express the final result in scientific notation:

1. a) $(3 \cdot 10^4) \cdot (2 \cdot 10^5)$
2. b) $(4 \cdot 10^{-3}) \cdot (2 \cdot 10^{-4})$
3. c) $(5 \cdot 10^6) \cdot (3 \cdot 10^{-2})$

Carry out the following divisions and give the result in scientific notation:

1. a) $(8 \cdot 10^7) : (2 \cdot 10^3)$
2. b) $(9 \cdot 10^{-4}) : (3 \cdot 10^{-2})$
3. c) $(1,5 \cdot 10^5) : (3 \cdot 10^8)$

Solve the following combined operation respecting the order of operations and give the final result adjusted in scientific notation:

$\frac{(4,2 \cdot 10^5) \cdot (2 \cdot 10^{-3})}{3 \cdot 10^4}$

Calculate the following additions and subtractions that have the same exponent in the power of $10$:

1. a) $4,2 \cdot 10^5 + 3,1 \cdot 10^5$
2. b) $8,7 \cdot 10^{-3} - 2,5 \cdot 10^{-3}$
3. c) $5,5 \cdot 10^8 + 4,5 \cdot 10^8$

Calculate the following additions and subtractions with different exponents, first adjusting the powers of $10$ so that they match. Express the result in scientific notation:

1. a) $3,5 \cdot 10^6 + 4,2 \cdot 10^5$
2. b) $7,1 \cdot 10^{-4} - 5 \cdot 10^{-5}$

Carry out the following combined operation with additions and subtractions in the numerator and denominator. Express the final result strictly in scientific notation:

$\frac{3,6 \cdot 10^{12} + 4,1 \cdot 10^{11}}{2 \cdot 10^{-3} - 5 \cdot 10^{-4}}$

In a microbiology experiment, a colony of bacteria initially contains $5 \cdot 10^4$ individuals and its population is observed to double every day. If the growth follows the model $P = P_0 \cdot 2^t$ (where $t$ is the number of days), how many bacteria will there be after $10$ days? Express the final result in scientific notation. (Helpful fact: $2^{10} = 1024$).

To make astronomical estimates, the Earth can be approximated as a perfect sphere with a radius of $R = 6,37 \cdot 10^3$ miles. Knowing that the volume of a sphere is calculated using the formula $V = \frac{4}{3}\pi R^3$, calculate the volume of the Earth in $km^3$. Express the final result in scientific notation, rounding to two decimal places. (Use $\pi \approx 3,14$).

The kinetic energy ($E_c$) of a moving body is calculated with the expression $E_c = \frac{1}{2} m v^2$, where $m$ is the mass in lb and $v$ is the speed in ft/s. Calculate the kinetic energy of an asteroid of mass $m = 4,5 \cdot 10^{14}$ lb traveling through space at a speed of $v = 2 \cdot 10^4$ ft/s. Express the result in Joules (J) using scientific notation.

Newton's Law of Universal Gravitation states that the force of attraction between two bodies is given by $F = G \frac{m_1 \cdot m_2}{d^2}$.

Knowing that the gravitational constant is $G = 6,67 \cdot 10^{-11}$ $N \cdot m^2/kg^2$, the mass of the Earth is $m_1 = 5,97 \cdot 10^{24}$ lb, the mass of the Moon is $m_2 = 7,35 \cdot 10^{22}$ lb and the mean distance between their centers is $d = 3,84 \cdot 10^8$ ft, calculate the force of gravity $F$ in Newtons ($N$). Express the result in scientific notation, rounding to two decimal places.

A human red blood cell has an approximate diameter of $7,5 \cdot 10^{-6}$ feet. If in a laboratory experiment we managed to align $4 \cdot 10^6$ red blood cells one after another, what length in feet would the line reach? Express the result in scientific notation and in standard decimal format.

The mean distance from the Earth to the Sun is about $1,5 \cdot 10^8$ miles and the speed of light in a vacuum is $3 \cdot 10^5$ miles/s. Using the formula for uniform rectilinear motion ($t = \frac{d}{v}$), how many seconds does it take for sunlight to reach the Earth? Express the result in scientific notation and then convert it to minutes.

The Gross Domestic Product (GDP) of a European country amounts to $\$1,2 \cdot 10^{12}$. If the total population of the country is $4 \cdot 10^7$ inhabitants, what is the GDP per capita (dollars per inhabitant)? Express the result in scientific notation and in standard dollars.

Set up and solve the following linear equation in which the coefficients are expressed in scientific notation. Solve for the unknown $x$ and express its value in scientific notation:

$(2,5 \cdot 10^4) \cdot x + 3 \cdot 10^5 = 1,8 \cdot 10^6$

A glass contains exactly $18$ grams of pure water ($H_2O$). In chemistry, it is known that $18$ grams are equivalent to $1$ mol of water and that $1$ mol of any substance contains Avogadro's Number ($6,022 \cdot 10^{23}$) of molecules. From this data, calculate the mass in kilograms of a single water molecule. Express the final result in scientific notation.

A research center needs to transfer a massive backup of $4,5$ Petabytes (PB) to a server in another city. The dedicated fiber optic connection allows data to be transferred at a constant speed of $3 \cdot 10^8$ bytes per second. Knowing that $1$ PB is equivalent to $10^{15}$ bytes, calculate how many full days (and fraction of a day if necessary) it will take to finish the transfer. Show the intermediate calculations in scientific notation.