Standard form

The video of a well-known Irish 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 $7.5 \times 10^{-6}$ 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.

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.

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) $9.99 \times 10^{12}$

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.

In a data processing centre 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 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 $1.2 \times 10^{13}$ 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 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 $t = \frac{d}{v}$).

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.

For an Economics report in 5th Year of secondary school (Leaving Certificate), Aoife has to compare and add two budget items that the Department of Education allocates to study grants: item $A = 4.8 \times 10^6$ euros and item $B = 7.5 \times 10^5$ euros. Aoife states the following in her draft:

"Since $7.5$ is greater than $4.8$, item $B$ has a larger financial endowment than item $A$. Furthermore, the total money allocated between both items is $(4.8 + 7.5) \times 10^{6+5} = 12.3 \times 10^{11} = 1.23 \times 10^{12}$ euros".

1. a) Identify the two conceptual errors made by Aoife (one in the comparison and another in the addition) and explain with reasoning why her deductions are mathematically false.
2. b) Correct Aoife's report: indicate which of the two items is actually larger, justifying your decision based on orders of magnitude, and calculate the exact value of the total sum $A + B$, expressing the result in rigorous scientific notation.

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.

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) $12.5 \times 10^{4} = 1.25 \times 10^{x}$

A group of researchers is analysing R&D investment by three world powers. The first invests $4.2 \times 10^{11} €$, the second $1.85 \times 10^{12} €$ and the third $9.7 \times 10^{10} €$.

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.

Given the following diagram representing a network of underground fibre-optic cables, the distance between inspection hatches $A$ and $B$ is given by the expression $d = \sqrt{5 \times 10^{-4}} + \sqrt{2 \times 10^{-5}}$ kilometres.

1. a) Simplify the expression for distance $d$ by taking factors out of the radicals and performing the operations to leave a single radical (or the simplest possible sum).

1. b) If a repair requires a cable whose length is exactly three times $d$, calculate this length and express it in metres using standard form to four 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.

The golden ratio, $\phi = \frac{1 + \sqrt{5}}{2}$, is present in the design of many modern mobile application interfaces to achieve visual harmony. Consider a smartphone screen following this ratio, where the width measures $w = 6.8 \times 10^{-2}$ metres.

1. a) Calculate the area of the screen in square millimetres ($mm^2$) knowing that the ratio between the height ($h$) and the width ($w$) is exactly $\phi$. Use the radical expression of $\phi$ for the calculation and approximate the final result in standard form to two decimal places.

1. b) If an error of $1 \times 10^{-4}$ metres in width occurs when manufacturing the glass (the height remaining unchanged), how does this affect the relative error of the area?

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 $7.5 \times 10^{-4} \text{ cm}$.

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 $2.5 \times 10^{13}$ cells), what distance in kilometres would they cover? Express the result in standard form.

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?

According to the annual reports of the Central Statistics Office (CSO), Ireland received approximately $8.4 \times 10^7$ international tourists in one year. Each visitor had an estimated average expenditure of $1.25 \times 10^3$ euros during their holiday in the country.

1. a) Without doing any calculations on paper or using a calculator, mentally estimate the order of magnitude (the power of $10$) that the overall tourist expenditure in euros should have. Clearly explain the reasoning you followed to establish that order.
2. b) Now perform the exact calculation of the total expenditure by multiplying both quantities and express the final result in correct scientific notation.
3. c) Check whether the calculated result matches your prior estimate and explain why analyzing the order of magnitude in advance allows you to quickly detect calculation errors when typing into a scientific calculator.

An auditing consultancy analyzes the accounts of an e-commerce company. It is known that the annual net profit, $B$, follows the formula $B = I - C$, where $I$ is the gross revenue and $C$ the total operating costs. In turn, the total costs result from making $N$ deliveries with a fixed unit logistics cost of $c_u = 3.5 \times 10^1$ euros per shipment (that is, $C = N \times c_u$). The following final figures are available:

• Gross revenue: $I = 4.2 \times 10^8$ euros.

• Net profit obtained: $B = 3.85 \times 10^8$ euros.

1. a) State the corresponding equation and calculate the value of the total operating costs $C$. Express the result in correct scientific notation.
2. b) Calculate the exact number of shipments $N$ made by the company during that financial year. Write $N$ in scientific notation, clearly identifying its mantissa and its order of magnitude.
3. c) If, looking ahead to the following year, revenue $I$ remained identical but total costs $C$ increased by exactly one order of magnitude, reason by analyzing the exponents whether the company would remain profitable or enter into losses, without needing to calculate the full subtraction.

A social network data centre 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 centre processes in a leap year.

1. b) If the centre 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 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.