Exercises: Standard form

Express the following integers and terminating decimals in standard form:

1. a) $450\,000\,000$
2. b) $32\,500$
3. c) $8\,400\,000\,000\,000$

Express the following small decimal numbers in standard form:

1. a) $0,0000045$
2. b) $0,00000000128$
3. c) $0,003$

Write out in full the following numbers given in standard form:

1. a) $3,14 \cdot 10^6$
2. b) $8,05 \cdot 10^9$
3. c) $1,002 \cdot 10^4$

Write the following numbers given in standard form in decimal form:

1. a) $5,2 \cdot 10^{-5}$
2. b) $1,07 \cdot 10^{-8}$
3. c) $9,9 \cdot 10^{-3}$

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

1. a) $(3 \cdot 10^4) \cdot (2 \cdot 10^5)$
2. b) $(4 \cdot 10^{-3}) \cdot (2 \cdot 10^8)$

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

1. a) $(8 \cdot 10^7) : (2 \cdot 10^3)$
2. b) $(9 \cdot 10^{-4}) : (3 \cdot 10^{-6})$

Calculate the following additions and subtractions that have the same exponent, giving the answer in standard form:

1. a) $4,2 \cdot 10^5 + 3,5 \cdot 10^5$
2. b) $8,7 \cdot 10^{-4} - 5,1 \cdot 10^{-4}$

Carry out the following operations by first adjusting the exponents so that they are equal, and express the final answer in standard form:

1. a) $3,5 \cdot 10^6 + 4,2 \cdot 10^5$
2. b) $9,1 \cdot 10^{-3} - 2,5 \cdot 10^{-4}$

Marcos has to calculate the combined mass of two telecommunications satellites placed into orbit. Satellite A has a mass of $4,5 \cdot 10^3\text{ kg}$ and satellite B has a mass of $8 \cdot 10^2\text{ kg}$. On his exam, he writes the following procedure:

1. a) Identify and explain the two conceptual errors Marcos made when working with powers and scientific notation.
2. b) Argue why it does not make physical sense for the total mass of two satellites (one of $4500\text{ kg}$ and the other of $800\text{ kg}$) to result in more than one million kilograms.
3. c) Perform the addition correctly step by step and express the final result in scientific notation.

According to data from the National Statistics Institute (INE), the resident population in Spain is approximately $4,8 \cdot 10^7$ people. It is estimated that the average daily water consumption in Spanish households is about $130\text{ litres}$ per inhabitant.

1. a) Express the daily water consumption per person in cubic metres using scientific notation (remember that $1\text{ m}^3 = 1000\text{ litres}$).
2. b) Model the total water consumption of all households in Spain during a full year ($365\text{ days}$). Express the result in cubic metres in scientific notation.
3. c) The capacity of the La Serena reservoir (Badajoz), the largest in Spain, is about $3,2 \cdot 10^9\text{ m}^3$. If it were at $100\,\%$ capacity, would it be enough to supply the annual domestic consumption of the entire country? Justify your answer by comparing the orders of magnitude and the coefficients.

The mean distance from the Earth to the Sun is about $149\,600\,000$ km and the speed of light in a vacuum is approximately $300\,000$ km/s. Express both quantities in standard form and calculate how many seconds it takes for light from the Sun to reach the Earth.

The size of a red blood cell is approximately $7,5 \cdot 10^{-6}$ m, while that of a flu virus is about $1,2 \cdot 10^{-7}$ m. How many flu viruses lined up end to end would be needed to equal the diameter of a red blood cell?

A storage unit in a data server has a capacity of $4$ terabytes ($4 \cdot 10^{12}$ bytes). If a text document occupies an average of $2,5 \cdot 10^4$ bytes, how many documents can be stored on the server? Express the answer in standard form.

Carry out the following calculation and express the final answer in standard form, adjusting the coefficient if necessary:

$\frac{(4,5 \cdot 10^{-5}) \cdot (6 \cdot 10^{12})}{3 \cdot 10^4}$

Order the following numbers from smallest to largest, having first expressed them in standard form (with a single non-zero digit before the decimal point):

$A = 0,0034 \cdot 10^8$

$B = 450 \cdot 10^3$

$C = 3,2 \cdot 10^5$

$D = 0,05 \cdot 10^7$

The mass of an oxygen atom is approximately $2,66 \cdot 10^{-23}$ g. Given that a drop of water contains about $1,67 \cdot 10^{21}$ oxygen atoms, calculate the total mass of oxygen present in the drop, expressing the answer in standard form.

In the year 2023, the Gross Domestic Product (GDP) of a country was $1,45 \cdot 10^{12}$ euros and its population was $4,7 \cdot 10^7$ inhabitants. Calculate the GDP per capita (euros per inhabitant), expressing the answer in standard form to two decimal places.

Calculate the following power by applying the relevant laws of indices and express the final answer strictly in standard form:

$(5 \cdot 10^{-4})^3$

To calculate how many times faster the speed of light in a vacuum ($c \approx 3 \cdot 10^8\text{ m/s}$) is than the speed of sound in air ($v \approx 3,4 \cdot 10^2\text{ m/s}$), two 4th-year ESO students debate which approach to take:

• Carla's strategy: First convert both numbers to ordinary decimal notation ($300\,000\,000$ and $340$) and then carry out the division by hand or with a calculator.

• David's strategy: Keep the scientific notation, directly divide the mantissas ($3 : 3,4$) and subtract the powers of ten ($10^{8-2}$), adjusting the mantissa at the end.

1. a) Explain the advantages of David's strategy over Carla's if you had to perform the calculation with pencil and paper without making zero-counting errors.
2. b) If you had to calculate the ratio between the mass of a proton ($1,67 \cdot 10^{-27}\text{ kg}$) and that of an electron ($9,11 \cdot 10^{-31}\text{ kg}$), would Carla's strategy still be feasible? Justify why scientific notation is the standard method in applied sciences.
3. c) Apply David's strategy to calculate how many times faster sound is than a turtle walking at $1,7 \cdot 10^{-1}\text{ m/s}$. Express the final result in scientific notation and in ordinary decimal notation.

According to the State Meteorological Agency (AEMET), the peninsular area of Spain is about $4,9 \cdot 10^5\text{ km}^2$. During a major autumn storm, an average rainfall of $20\text{ mm}$ of rain was recorded across the entire peninsular territory (remember that $1\text{ mm} = 1\text{ litre/m}^2$, and that $1\text{ litre} = 10^{-3}\text{ m}^3$).

1. a) Estimate before calculating: Knowing that $1\text{ km}^2 = 10^6\text{ m}^2$, make a quick mental estimate of the order of magnitude (the power of ten) of the total volume of fallen water in cubic metres ($10^9$, $10^{10}$, $10^{11}\dots$) and write down your reasoned prediction.
2. b) Calculate: Convert the peninsular area to square metres in scientific notation and calculate the exact total volume of precipitated water in cubic metres. Express the result in scientific notation.
3. c) Check: Compare the exact result with your initial estimate. Knowing that water consumption for all uses in Spain (agricultural, industrial, and urban) is around $3 \cdot 10^{10}\text{ m}^3$ per year, assess whether it is reasonable that a single heavy storm provides an amount on the order of $30\,\%$ of the country's entire annual consumption.

The law of universal gravitation states that the gravitational force between two masses $m_1$ and $m_2$ separated by a distance $d$ is $F = G \cdot \frac{m_1 \cdot m_2}{d^2}$, where $G = 6,67 \cdot 10^{-11} \text{ N}\cdot\text{m}^2/\text{kg}^2$. Calculate the force of attraction between the Earth ($m_1 = 5,97 \cdot 10^{24}$ kg) and the Moon ($m_2 = 7,35 \cdot 10^{22}$ kg), given that the mean distance between their centres is $3,84 \cdot 10^8$ m. Give the answer in standard form to three significant figures.

Carry out the following calculation, paying attention to the order of operations and adjusting the exponents appropriately before adding. Express the final answer in standard form:

$\frac{3,2 \cdot 10^8 + 4,5 \cdot 10^7}{(2 \cdot 10^{-3})^2}$

The estimated age of the universe is about $13\,800$ million years. If a year has approximately $3,15 \cdot 10^7$ seconds, calculate the age of the universe in seconds, expressed in standard form. If the mean lifetime of an unstable subatomic particle is $2,5 \cdot 10^{-20}$ seconds, how many lifetimes of this particle fit sequentially into the age of the universe?

A laser beam emits pulses of light with a total energy of $3,5 \cdot 10^{-3}$ joules per pulse. If the energy of a single photon of this light is $3,14 \cdot 10^{-19}$ joules, how many photons are in each pulse? If the laser emits $50$ pulses per second, how many photons does it emit in one hour of continuous operation? Express both answers in standard form.

When measuring the distance between two distant galaxies, a value of $24\,560\,000\,000\,000\,000\,000$ km is obtained. Express this quantity in standard form rounded to three significant figures (that is, two decimal places). Next, calculate the absolute error and the relative error made when making this approximation.

Find the value of the exponent $x$ (an integer) and the coefficient $y$ (with $1 \le y < 10$) so that the following equality holds:

$(y \cdot 10^x)^3 = 1,25 \cdot 10^{17}$

It is desired to coat a sphere of radius $R = 2 \cdot 10^{-2}$ m with a layer of gold that is a single atom thick. The radius of a gold atom is approximately $1,44 \cdot 10^{-10}$ m. Assuming that each atom occupies an area on the surface equal to a square whose side length is equal to its diameter, estimate how many gold atoms will be required to completely cover the sphere. (Recall that the surface area of a sphere is $A = 4\pi R^2$).

Let a number in standard form be $N = a \cdot 10^n$, where $1 \le a < 10$ and $n$ is an odd integer. When calculating the square root of $N$, the expression $\sqrt{a \cdot 10^n}$ is obtained. Describe the mathematical procedure so that the final result is correctly written in standard form $b \cdot 10^m$, specifying the values of $b$ and $m$ in terms of $a$ and $n$. Apply your reasoning to calculate the exact value of $\sqrt{4,9 \cdot 10^{15}}$ in standard form.

In a digital security system, two numerical keys expressed in scientific notation are generated, $X = a \cdot 10^p$ and $Y = b \cdot 10^q$, with $1 \le a, b < 10$ and $p, q$ integers. To regain access to the system, the following conditions are known:

1. The mantissas $a$ and $b$ are integers greater than 1.

2. The product of the two keys is exactly $X \cdot Y = 1,4 \cdot 10^7$.

3. The quotient $\frac{Y}{X}$, expressed in scientific notation, has an order of magnitude of $10^{13}$ (that is, its exponent is $13$).

4. The exponent $p$ is strictly negative and the exponent $q$ is strictly positive.

1. a) Reason why the product of the mantissas $a \cdot b$ must necessarily be equal to $14$ and deduce what the values of the integers $a$ and $b$ must be.
2. b) From the order of magnitude of the quotient and the product, set up a system of equations with the exponents $p$ and $q$, solve it, and write the numbers $X$ and $Y$ uniquely.
3. c) Verify that your solution satisfies all the conditions in the statement by explicitly calculating both the product $X \cdot Y$ and the division $\frac{Y}{X}$.

Red Eléctrica de España (REE) records an annual peninsular electricity demand of approximately $2,4 \cdot 10^{11}\text{ kWh}$. As part of ecological transition plans, the goal is to cover $15\,\%$ of that national demand exclusively through solar installations on the rooftops of single-family homes. The following technical data are available:

• Each domestic installation generates an average of $5 \cdot 10^3\text{ kWh}$ per year.

• Each installation requires a collection surface area of $25\text{ m}^2$ of panels.

• Each kilowatt-hour ($1\text{ kWh}$) generated with solar energy saves the emission of $2,6 \cdot 10^{-1}\text{ kg}$ of $\text{CO}_2$ compared to generation with thermal power plants.

1. a) Model the amount of energy in $\text{kWh}$ corresponding to that $15\,\%$ of the demand and calculate how many homes with solar panels would be needed in total (express both results in scientific notation).
2. b) Calculate how many tonnes of $\text{CO}_2$ would be prevented from being emitted into the atmosphere each year with this plan ($1\text{ tonne} = 10^3\text{ kg}$).
3. c) Calculate the total surface area of solar panels needed in square kilometres ($1\text{ km}^2 = 10^6\text{ m}^2$). Knowing that the municipality of Madrid has an area of about $6 \cdot 10^2\text{ km}^2$, compare both surface areas and reflect in a reasoned manner on whether this rollout is feasible distributed across the rooftops of the millions of buildings in the country.