Exercises: Standard form

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 decimals:

1. a) $3.4 \times 10^5$
2. b) $1.89 \times 10^{-4}$
3. c) $7 \times 10^8$
4. d) $5.02 \times 10^{-6}$

State which of the following numbers are not correctly written in standard form and write them correctly:

1. a) $45.2 \times 10^3$
2. b) $0.8 \times 10^{-2}$
3. c) $3.1 \times 10^5$
4. d) $12 \times 10^{-4}$

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

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

Carry out the following divisions and give the answer in standard form:

1. a) $(8 \times 10^7) : (2 \times 10^3)$
2. b) $(9 \times 10^{-4}) : (3 \times 10^{-2})$
3. c) $(1.5 \times 10^5) : (3 \times 10^8)$

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

1. a) $4.2 \times 10^5 + 3.1 \times 10^5$
2. b) $8.7 \times 10^{-3} - 2.5 \times 10^{-3}$
3. c) $5.5 \times 10^8 + 4.5 \times 10^8$

Order the following numbers written in standard form from smallest to largest:

$A = 3.2 \times 10^4$

$B = 4.1 \times 10^{-3}$

$C = 1.5 \times 10^4$

$D = 8.9 \times 10^{-4}$

$E = 2 \times 10^5$

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

1. a) $4.8 \times 10^6$
2. b) $1.1 \times 10^{-5}$
3. c) $9.2 \times 10^{12}$
4. d) $2.5 \times 10^{-8}$

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

1. a) $3.5 \times 10^6 + 4.2 \times 10^5$
2. b) $7.1 \times 10^{-4} - 5 \times 10^{-5}$

A human red blood cell has an approximate diameter of $7.5 \times 10^{-6}$ meters. If in a laboratory experiment we managed to align $4 \times 10^6$ red blood cells end to end, what length in meters would the row reach? Express the answer in scientific notation and in ordinary decimal format.

The mean distance from the Earth to the Sun is about $1.5 \times 10^8$ km and the speed of light in a vacuum is $3 \times 10^5$ km/s. Using the formula for uniform motion in a straight line ($t = \frac{d}{v}$), how many seconds does it take for sunlight to reach the Earth? Express the answer 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 occupies an average of $4 \times 10^6$ bytes, what is the maximum number of photographs that can be stored on this hard drive?

Note: Assume that $1$ TB is equivalent to $10^{12}$ bytes.

The Gross Domestic Product (GDP) of a European country amounts to $1.2 \times 10^{12}$ pesos. If the total population of the country is $4 \times 10^7$ inhabitants, what is the GDP per capita (pesos per inhabitant)? Express the answer in scientific notation and in ordinary pesos.

In particle physics, it is known that the mass of an electron is approximately $9.1 \times 10^{-31}$ kg and the mass of a proton is about $1.67 \times 10^{-27}$ kg. Calculate how many times more massive a proton is than an electron. Express the solution in standard form to two decimal places.

Solve the following combined operation respecting the order of operations and give the final result in standard form:

$\frac{(4.2 \times 10^5) \times (2 \times 10^{-3})}{3 \times 10^4}$

A reservoir for human consumption contains $4.5 \times 10^3$ 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 scientific notation. If the city it supplies consumes $5 \times 10^7$ liters of water daily, for how many days are there sufficient reserves?

To make astronomical estimates, the Earth can be approximated as a perfect sphere with a radius of $R = 6.37 \times 10^3$ km. 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 rounded to two decimal places. (Use $\pi \approx 3.14$).

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

$\frac{3.6 \times 10^{12} + 4.1 \times 10^{11}}{2 \times 10^{-3} - 5 \times 10^{-4}}$

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

$(2.5 \times 10^4) \times x + 3 \times 10^5 = 1.8 \times 10^6$

The kinetic energy ($E_c$) of a moving body is calculated using 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 $m = 4.5 \times 10^{14}$ kg travelling through space at a speed of $v = 2 \times 10^4$ m/s. Express the answer in Joules (J) using standard form.

A newly uploaded video has $5 \times 10^4$ views and it is observed that this figure doubles every day. If the growth follows the model $V = V_0 \times 2^t$ (where $t$ is the number of days), how many views 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 gravitational force between two bodies is given by $F = G \frac{m_1 \times m_2}{d^2}$.

Knowing that the gravitational constant is $G = 6.67 \times 10^{-11}$ $N \times m^2/kg^2$, the mass of the Earth is $m_1 = 5.97 \times 10^{24}$ kg, the mass of the Moon is $m_2 = 7.35 \times 10^{22}$ kg and the mean distance between their centers is $d = 3.84 \times 10^8$ m, calculate the gravitational force $F$ in Newtons ($N$). Express the result in standard form rounded 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 ($6.022 \times 10^{23}$) of molecules. From these data, calculate the mass in kilograms of a single molecule of water. 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 $3 \times 10^8$ 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 complete the transfer. Show the intermediate calculations in scientific notation.