Scientific Notation

The video of a well-known Mexican streamer has reached $45,200,000$ views in just one week. Express this number of views in scientific notation and state its mantissa and its order of magnitude.

In the biology laboratory, a group of students observes that the diameter of a red blood cell is approximately $7.5 \times 10^{-6}$ meters. Express this measurement in decimal form (ordinary number notation) and explain in which direction 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) $0.45 \times 10^{8}$
2. b) $1.02 \times 10^{-5}$
3. c) $12.5 \times 10^{3}$
4. d) $9.99 \times 10^{12}$

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

1. a) $345,000 = 3.45 \times 10^{x}$
2. b) $0.00007 = 7 \times 10^{x}$
3. c) $12.5 \times 10^{4} = 1.25 \times 10^{x}$

Look at the following diagram representing the movement of the decimal point to convert the number $0.000042$ to standard form. Complete the boxes with the value of the coefficient (mantissa) and the corresponding exponent.

In microelectronics, the thickness of an insulating layer in a processor is $4.5$ nanometers ($1 \text{ nm} = 10^{-9} \text{ m}$). If the processor has a total surface area of $1.2 \text{ cm}^2$:

1. a) Express the thickness of the layer in meters using scientific notation.

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

A technology company is comparing the size of three electronic components. The sizes are: $A = 3.2 \times 10^{-4}$ mm, $B = 1.5 \times 10^{-3}$ mm and $C = 8.9 \times 10^{-4}$ mm. Order the components from smallest to largest based on their exponents and mantissas.

The mean distance from the Earth to the Sun is approximately $1.5 \times 10^{8}$ km. If light travels at a speed of $3 \times 10^{5}$ km/s, calculate the time it takes for sunlight to reach the Earth, carrying out the operation directly in standard form (remember that $t = \frac{d}{v}$).

A data server stores music files. Each song occupies an average of $4.5 \times 10^{6}$ bytes. If the server has $2 \times 10^{4}$ songs stored, how many total bytes of music are on the server? Express the final result in standard form.

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

Calculate the value of the following expression, simplifying the result as far as possible, and express it in standard form to three significant figures:

Next, bound the absolute error made when rounding the value of $L$ to the first decimal place of its mantissa.

In a data processing center of a well-known social network, it is estimated that $2.5 \times 10^6$ megabytes (MB) of new information are generated every second.

1. a) Express in scientific notation 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 $1.2 \times 10^{13}$ 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 mass of a suspended dust particle is approximately $7.25 \times 10^{-10}$ kg. After one week of operation, an air purifier has captured a total of $4.35 \times 10^{-2}$ 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.

The mass of a laboratory object has been measured, giving a value of $3.45 \times 10^{-3}$ grams, although the true value is $3.458 \times 10^{-3}$ grams.

1. a) Calculate the absolute error made in the measurement.

1. b) Calculate the relative error and express it as a percentage. Is it considered a precise measurement if the acceptable margin of error is $0.5\%$? Justify your answer.

Represent on the number line the set of numbers satisfying the following "industrial tolerance" condition for a precision part:

1. a) Express the set in interval form and as an inequality without modulus.

1. b) Is the number $0.00435$ an acceptable value for this part? Justify your answer by checking whether it belongs to the set.

You can do it on paper and send a photo with your answer.

In a nanotechnology laboratory, the mass of a new synthetic particle is being analysed, estimated at $m = 4.25 \times 10^{-22}$ grams. However, the measuring instrument has an absolute error bound of $5 \times 10^{-25}$ grams. On the other hand, an astronomer measures the mass of a nearby asteroid as $M = 7.8 \times 10^{15}$ 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 $1.5 \times 10^{24}$ identical particles were combined, what would be the resulting total mass? Consider both the estimated value and the accumulated absolute error.

A red blood cell has an approximately circular shape with a diameter of $7.5 \times 10^{-4} \text{ cm}$.

1. a) Calculate the circumference of the cell in meters. (Use $\pi \approx 3.1416$).

1. b) If we could line up end to end all the red blood cells of an adult (approximately $2.5 \times 10^{13}$ cells), what distance in kilometers would they cover? Express the result in scientific notation.

The mean distance from the Earth to the Sun is $1.496 \times 10^8 \text{ km}$. The speed of light is $3 \times 10^8 \text{ m/s}$.

1. a) Calculate how long, in minutes and seconds, it takes for sunlight to reach the Earth.

1. b) If a spacecraft travels at a constant speed of $5.5 \times 10^4 \text{ km/h}$, how many days would it take to complete the round trip?

A social network data center processes approximately $2.5 \times 10^6$ queries per second. Each query requires the movement of $1.2 \times 10^4$ bytes of information.

1. a) Calculate how many terabytes ($1 \text{ TB} = 10^{12} \text{ bytes}$) of information the center processes in a leap year.

1. b) If the center plans to increase its processing capacity by $15\%$ annually compounded, how much information (in bytes) will it process per second in a decade? Express the result in scientific notation.