Exercises: 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.

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.

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.

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.

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.

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.

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 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.

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.

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?

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.

The Bank of Spain recorded the national public debt at an approximate figure of $1.58 \times 10^{12}$ euros at the end of a year. During that same period, the census showed a total population of $4.8 \times 10^7$ people in Spain, although the employed population (active workers generating social insurance contributions and direct tax revenues) was only $2.1 \times 10^7$ people.

1. a) Model the "per capita debt" by dividing the total debt by the total number of inhabitants. Express the quotient in scientific notation with three significant figures and then write it in ordinary decimal format (euros per inhabitant).
2. b) Model the "debt per active worker", dividing the total debt exclusively by the employed population. Express the result in scientific notation with three significant figures.
3. c) Deduce from the calculated mantissas and exponents how many times larger the theoretical burden borne by an active worker is compared to the average of the general population. Explain why the use of scientific notation simplifies the comparison of macroeconomic magnitudes compared to traditional numerical notation.

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.

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.

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.

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 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?

In a global reforestation campaign, it is estimated that each planted tree absorbs an average of $2.2 \times 10^1$ kg of $CO_2$ per year. The organising platform claims to have offset $4.5 \times 10^8$ tonnes of $CO_2$ over the last 5 years ($1 \text{ tonelada} = 10^3 \text{ kg}$).

1. a) Assuming that the number of trees planted has been constant and that absorption is cumulative (trees planted in the first year absorb over all five years, those planted in the second year over four years, etc.), calculate how many trees were planted each year.

1. b) If there is an error in the absorption measurement of $\pm 5\%$ per tree, calculate the interval for the absolute error of the total number of tonnes offset.

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.

The world population is projected to reach $9.7 \times 10^9$ people in the year 2050. If the total land surface area is approximately $1.48 \times 10^8 \text{ km}^2$, of which only $25\%$ is habitable:

1. a) Calculate the population density (inhabitants per $km^2$) in the habitable areas for the year 2050.

1. b) If each person consumes an average of $1.3 \times 10^6$ kilocalories per year, how much total food energy (in kcal) will humanity need in a decade (10 years), assuming the population remains constant at the 2050 figure? Give the result in standard form.

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.

An investment firm manages a portfolio valued at $F = 5.2 \times 10^{10}$ euros. At the end of the fiscal year, it must incorporate two adjustments: the collection of interest accrued by an auxiliary account, which amounts to $I = 6.8 \times 10^4$ euros, and the payment of a regulatory fine in the amount of $S = 4.1 \times 10^7$ euros. The final adjusted net worth follows the expression $P = F + I - S$.

To present the report to the board of directors with an accuracy of four significant figures, two economists propose different approaches:

• Method 1 (Full decimal expansion): Expand all the zeros of each of the three numbers, arrange them in columns according to place values, and perform traditional addition and subtraction in standard positional notation, then convert back to scientific notation.

• Method 2 (Factorization of orders of magnitude): Factor out the predominant power ($10^{10}$), express the secondary amounts as mantissas associated with that exponent ($I = 0.0000068 \times 10^{10}$ and $S = 0.0041 \times 10^{10}$), and analyze the relevance of each term before rounding to four significant figures.

1. a) Apply Method 2 showing the steps of the operations and obtain the final value of $P$ with four significant figures in scientific notation. Explain what happens to the amount $I$ during this process and justify whether it is mathematically sound to ignore it.
2. b) Analyze the advantages and disadvantages of both methods based on three criteria: risk of error due to omission/addition of zeros, computational time cost, and suitability for floating-point computer systems.
3. c) Justify with reasoning which of the two methods demonstrates a deeper understanding of the structure of real numbers and the handling of orders of magnitude.

In a study on the management of water resources in the Shannon river basin, the total reservoir capacity is set at $C = 8.0 \times 10^9 \text{ m}^3$. At the beginning of the agricultural year, the stored water is at $25\%$ of its total maximum capacity.

Throughout the $365$ days of the year, three water inflows and withdrawals are expected:

1. Inflows from rainfall and river runoff: a constant flow rate that contributes $1.5 \times 10^6 \text{ m}^3$ of water per day.

2. Losses due to natural evaporation from the surface of the reservoirs: an accumulated total over the year of $1.25 \times 10^8 \text{ m}^3$.

3. Extraction for human and urban consumption, and agricultural irrigation: an annual total volume of $V_E = k \times 10^8 \text{ m}^3$ is extracted, where $k$ is a parameter satisfying $1 \le k < 10$.

1. a) Express in rigorous scientific notation: the initial volume of stored water and the total annual volume contributed by rainfall.
2. b) To prevent severe restrictions in cities, authorities determine that the final volume of reservoir water at the end of the $365$ days cannot fall below an emergency reserve of $1.8 \times 10^9 \text{ m}^3$. State an inequality with all terms expressed in scientific notation and find the maximum allowable value for the mantissa $k$.
3. c) Suppose that an extreme drought reduces the rainfall for the entire year by exactly one order of magnitude compared to the initial forecast. Prove mathematically whether it would be feasible to maintain the emergency reserve of $1.8 \times 10^9 \text{ m}^3$ even in the extreme case of completely suspending human consumption and irrigation ($k = 0$).