Scientific Notation

Indicate which of the following numbers are not correctly written in standard form 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 standard form:

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

Express the following numbers given in standard form as ordinary decimal numbers:

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 standard form from smallest to largest:

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

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

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

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

$E = 2 \cdot 10^5$

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

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 standard form:

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 standard form:

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 standard form:

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

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 standard form:

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 standard form:

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

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

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

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

An external hard drive has a storage capacity of $2$ Terabytes (TB). If a high-resolution photograph takes up an average of $4 \cdot 10^6$ bytes, what is the maximum number of photographs that can be stored on this hard drive?

Fact: Consider that $1$ TB is equivalent to $10^{12}$ bytes.

The Gross Domestic Product (GDP) of a 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 standard form and in ordinary dollars.

In particle physics, it is known that the mass of an electron is approximately $9,1 \cdot 10^{-31}$ kg and the mass of a proton is about $1,67 \cdot 10^{-27}$ kg. Calculate how many times more massive a proton is than an electron. Express the solution in standard form to two decimal places.

A water reservoir for human consumption contains $4,5 \cdot 10^3$ cubic hectometers ($hm^3$). Knowing that $1$ $hm^3$ is equivalent to $10^9$ liters, express the total capacity of the reservoir in liters using standard form. If the city it supplies consumes $5 \cdot 10^7$ liters of water daily, for how many days are there sufficient reserves?

To make astronomical estimates, the Earth can be approximated to 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 standard form, 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 kg and $v$ is the speed in m/s. Calculate the kinetic energy of an asteroid of mass $m = 4,5 \cdot 10^{14}$ kg traveling through space at a speed of $v = 2 \cdot 10^4$ m/s. Express the result in Joules (J) using standard form.

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 standard form. (Helpful fact: $2^{10} = 1024$).

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 $m_1 = 5,97 \cdot 10^{24}$ kg, the mass of the Moon is $m_2 = 7,35 \cdot 10^{22}$ kg and the mean distance between their centers is $d = 3,84 \cdot 10^8$ m, calculate the force of gravity $F$ in Newtons ($N$). Express the result in standard form, rounding to two decimal places.

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 standard form.

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 standard form.