Scientific notation

The video of a well-known Spanish streamer has reached $45.200.000$ views in just one week. Express this number of views using standard form and state its mantissa and its order of magnitude.

In the Biology lab, a group of students observes that the diameter of a red blood cell is approximately $7,5 \cdot 10^{-6}$ meters. The figure shows the red blood cell with its diameter measurement.

Express this measurement in decimal form (standard notation) and explain which way the decimal point moves and by how many places.

State which of the following numbers are correctly written in standard form. If any are not, briefly explain why:

1. a) $0,45 \cdot 10^{8}$
2. b) $1,02 \cdot 10^{-5}$
3. c) $12,5 \cdot 10^{3}$
4. d) $9,99 \cdot 10^{12}$

A technology company is comparing the size of three electronic components. The sizes are: $A = 3,2 \cdot 10^{-4}$ mm, $B = 1,5 \cdot 10^{-3}$ mm and $C = 8,9 \cdot 10^{-4}$ mm. Order the components from smallest to largest based on their exponents and mantissas.

In a data processing centre of a well-known social network, it is estimated that $2,5 \cdot 10^6$ megabytes (MB) of new information are generated every second.

1. a) Express in standard form how many gigabytes (GB) are generated in a full day (assume $1 \text{ GB} = 10^3 \text{ MB}$).

1. b) If each storage hard drive has a capacity of $1,2 \cdot 10^{13}$ bytes, how many hard drives would be needed to store the information of a leap year? (assume $1 \text{ MB} = 10^6 \text{ bytes}$). Give the result to three significant figures.

The mean distance from the Earth to the Sun is approximately $1,5 \cdot 10^{8}$ km. If light travels at a speed of $3 \cdot 10^{5}$ km/s, calculate the time it takes for sunlight to reach the Earth, carrying out the operation directly in standard form (remember that $t = \frac{d}{v}$).

The mass of a suspended dust particle is approximately $7,25 \cdot 10^{-10}$ kg. After one week of operation, an air purifier has captured a total of $4,35 \cdot 10^{-2}$ grams of these particles.

1. a) Calculate the total number of particles captured by the purifier.

1. b) If the purifier filter must be changed when it reaches an accumulated mass of $1,5 \text{ decigramos}$, how many more weeks can it operate before requiring maintenance? Express the calculations and the final result in standard form.

A data server stores music files. Each song occupies an average of $4,5 \cdot 10^{6}$ bytes. If the server has $2 \cdot 10^{4}$ songs stored, how many total bytes of music are on the server? Express the final result in standard form.

Determine the value of $x$ in each of the following equations so that equivalence in standard form is maintained:

1. a) $345.000 = 3,45 \cdot 10^{x}$
2. b) $0,00007 = 7 \cdot 10^{x}$
3. c) $12,5 \cdot 10^{4} = 1,25 \cdot 10^{x}$

A group of researchers is analysing R&D investment by three world powers. The first invests $4,2 \cdot 10^{11} €$, the second $1,85 \cdot 10^{12} €$ and the third $9,7 \cdot 10^{10} €$.

1. a) Calculate the total investment of the three powers combined, expressing the result in standard form.

1. b) What percentage of the total is represented by the investment of the power that contributes the least money? Round the percentage to two decimal places.

In microelectronics, the thickness of an insulating layer in a processor is $4,5$ nanometres ($1 \text{ nm} = 10^{-9} \text{ m}$). If the processor has a total surface area of $1,2 \text{ cm}^2$:

1. a) Express the thickness of the layer in metres using standard form.

1. b) Calculate the total volume of insulating material used in that processor in cubic metres ($m^3$). Take into account that the volume is the product of the surface area and the thickness.

Carry out the following combined operations and express the result in standard form, rounding to three significant figures:

A red blood cell has an approximately circular shape with a diameter of $7,5 \cdot 10^{-4} \text{ cm}$.

1. a) Calculate the circumference of the cell in metres. (Use $\pi \approx 3,1416$).

1. b) If we could line up end to end all the red blood cells of an adult (approximately $2,5 \cdot 10^{13}$ cells), what distance in kilometres would they cover? Express the result in standard form.

In a nanotechnology laboratory, the mass of a new synthetic particle is being analysed, estimated at $m = 4,25 \cdot 10^{-22}$ grams. However, the measuring instrument has an absolute error bound of $5 \cdot 10^{-25}$ grams. On the other hand, an astronomer measures the mass of a nearby asteroid as $M = 7,8 \cdot 10^{15}$ kilograms with a relative error of $0,01\%$.

Carry out the following tasks, expressing all results in standard form:

1. a) Determine the interval of possible values for the mass of the particle $m$ using neighbourhood notation $E(a, r)$.

1. b) Calculate the relative error bound made in measuring the particle and compare it with the relative error of the asteroid. Give reasons to justify which measurement is more precise.

1. c) If $1,5 \cdot 10^{24}$ identical particles were combined, what would be the resulting total mass? Consider both the estimated value and the accumulated absolute error.