Exercises: Standard form

Write the following decimal numbers in standard form:

1. a) $450 000 000$
2. b) $0,000000032$
3. c) $12 500$
4. d) $0,000105$

Express the following numbers given in standard form as ordinary decimals:

1. a) $3,4 \cdot 10^5$
2. b) $1,89 \cdot 10^{-4}$
3. c) $7 \cdot 10^8$
4. d) $5,02 \cdot 10^{-6}$

State which of the following numbers are not correctly written in standard form and write them correctly:

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

Perform the following multiplications by grouping the numbers and the powers of base $10$. Express the final result in standard form:

1. a) $(3 \cdot 10^4) \cdot (2 \cdot 10^5)$
2. b) $(4 \cdot 10^{-3}) \cdot (2 \cdot 10^{-4})$
3. c) $(5 \cdot 10^6) \cdot (3 \cdot 10^{-2})$

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

1. a) $(8 \cdot 10^7) : (2 \cdot 10^3)$
2. b) $(9 \cdot 10^{-4}) : (3 \cdot 10^{-2})$
3. c) $(1,5 \cdot 10^5) : (3 \cdot 10^8)$

Calculate the following additions and subtractions that have the same exponent in the power of $10$:

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

Order the following numbers written in standard form from smallest to largest:

$A = 3,2 \cdot 10^4$

$B = 4,1 \cdot 10^{-3}$

$C = 1,5 \cdot 10^4$

$D = 8,9 \cdot 10^{-4}$

$E = 2 \cdot 10^5$

Determine the order of magnitude (the closest power of $10$) of the following quantities expressed in standard form:

1. a) $4,8 \cdot 10^6$
2. b) $1,1 \cdot 10^{-5}$
3. c) $9,2 \cdot 10^{12}$
4. d) $2,5 \cdot 10^{-8}$

Marcos is solving a laboratory exercise where he must calculate the product of two quantities: $A = 4 \cdot 10^5$ and $B = 3,5 \cdot 10^3$. In his notebook he writes the following working:

1. a) Identify and explain the two conceptual errors Marcos made (one in the properties of powers and another when adjusting the significand to the definition of scientific notation).
2. b) Show the correct step-by-step procedure and write the final result in scientific notation.

The goal is to calculate how many times greater the average distance from the Earth to the Moon ($d_L = 3,84 \cdot 10^5\text{ km}$) is than the length of the Earth's equator ($L_E = 4,0 \cdot 10^4\text{ km}$).

1. a) Estimate: Before performing any exact calculation, round the significand of $d_L$ to the nearest integer and calculate mentally the approximate order of magnitude of the quotient $\frac{d_L}{L_E}$ (that is, approximately how many full trips around the equator a journey to the Moon equals).
2. b) Calculate: Now perform the division with the exact data by applying the properties of scientific notation.
3. c) Check: Compare the value obtained with your initial estimate and reason whether your result is consistent and physically plausible.

Calculate the following additions and subtractions with different exponents, first adjusting the powers of $10$ so that they match. Express the answer in standard form:

1. a) $3,5 \cdot 10^6 + 4,2 \cdot 10^5$
2. b) $7,1 \cdot 10^{-4} - 5 \cdot 10^{-5}$

A human red blood cell has an approximate diameter of $7,5 \cdot 10^{-6}$ metres. If in a laboratory experiment we managed to align $4 \cdot 10^6$ red blood cells end to end, what length in metres would the row reach? Express the answer in standard form and in ordinary decimal format.

The mean distance from the Earth to the Sun is about $1,5 \cdot 10^8$ km and the speed of light in a vacuum is $3 \cdot 10^5$ km/s. Using the formula for uniform motion in a straight line ($t = \frac{d}{v}$), how many seconds does it take for sunlight to reach the Earth? Express the answer in standard form and then convert it to minutes.

An external hard drive has a storage capacity of $2$ terabytes (TB). If a high-resolution photograph occupies an average of $4 \cdot 10^6$ bytes, what is the maximum number of photographs that can be stored on this hard drive?

Note: Assume that $1$ TB is equivalent to $10^{12}$ bytes.

The Gross Domestic Product (GDP) of a European country amounts to $1,2 \cdot 10^{12}$ euros. If the total population of the country is $4 \cdot 10^7$ inhabitants, what is the GDP per capita (euros per inhabitant)? Express the answer in standard form and in ordinary euros.

In particle physics, it is known that the mass of an electron is approximately $9,1 \cdot 10^{-31}$ kg and the mass of a proton is about $1,67 \cdot 10^{-27}$ kg. Calculate how many times more massive a proton is than an electron. Express the solution in standard form to two decimal places.

Solve the following combined operation respecting the order of operations and give the final result in standard form:

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

A reservoir for human consumption contains $4,5 \cdot 10^3$ cubic hectometres ($hm^3$). Knowing that $1$ $hm^3$ is equivalent to $10^9$ litres, express the total capacity of the reservoir in litres using standard form. If the city it supplies consumes $5 \cdot 10^7$ litres of water daily, for how many days are there sufficient reserves?

According to data from the National Statistics Institute (INE), the average water consumption in Spanish households is approximately $133\text{ liters}$ per person per day. Around $4,8 \cdot 10^7$ people live in Spain.

1. a) Express the consumption of one person in one day in cubic meters ($1\text{ m}^3 = 10^3\text{ liters}$) using scientific notation.
2. b) Calculate the total volume of water consumed in households across all of Spain during a non-leap year ($3,65 \cdot 10^2\text{ days}$). Express the result in cubic meters in scientific notation.
3. c) The Alarcón reservoir has an approximate maximum capacity of $1,1 \cdot 10^9\text{ m}^3$. How many reservoirs like this would be needed to supply Spain's annual domestic consumption? Round to the nearest tenth and briefly explain the importance of domestic water conservation in light of this volume.

To calculate the exact value of the expression:

three students propose different strategies:

• Strategy A (Elena): Convert the three terms to decimal notation with all their leading zeros, add and subtract them in columns, and finally convert the result to scientific notation.

• Strategy B (Bernat): Transform all terms so they share the power of the largest term ($10^{-4}$), factor out that power, operate on the significands, and readjust if necessary.

• Strategy C (Clara): Transform all terms to the power of the smallest exponent ($10^{-6}$), operate on the resulting integer significands, and then readjust to scientific notation.

1. a) Solve the operation following Strategy B and express the result in scientific notation.
2. b) Critically evaluate Strategy A: what mathematical and practical risks does it entail when working with very small or very large exponents?
3. c) Compare Strategy B and Strategy C: which of the two minimizes the probability of having to readjust the significand at the end so that it falls in the interval $[1, 10)$? Justify which one you would recommend using in general.

To make astronomical estimates, the Earth can be approximated as a perfect sphere with a radius of $R = 6,37 \cdot 10^3$ km. Knowing that the volume of a sphere is calculated using the formula $V = \frac{4}{3}\pi R^3$, calculate the volume of the Earth in $km^3$. Express the final result in standard form rounded to two decimal places. (Use $\pi \approx 3,14$).

Carry out the following combined operation with additions and subtractions in the numerator and denominator. Express the final result strictly in standard form:

$\frac{3,6 \cdot 10^{12} + 4,1 \cdot 10^{11}}{2 \cdot 10^{-3} - 5 \cdot 10^{-4}}$

Set up and solve the following linear equation in which the coefficients are expressed in standard form. Rearrange for the unknown $x$ and express its value in standard form:

$(2,5 \cdot 10^4) \cdot x + 3 \cdot 10^5 = 1,8 \cdot 10^6$

The kinetic energy ($E_c$) of a moving body is calculated using the expression $E_c = \frac{1}{2} m v^2$, where $m$ is the mass in kg and $v$ is the speed in m/s. Calculate the kinetic energy of an asteroid of mass $m = 4,5 \cdot 10^{14}$ kg travelling through space at a speed of $v = 2 \cdot 10^4$ m/s. Express the answer in Joules (J) using standard form.

A newly uploaded video has $5 \cdot 10^4$ views and it is observed that this figure doubles every day. If the growth follows the model $V = V_0 \cdot 2^t$ (where $t$ is the number of days), how many views will there be after $10$ days? Express the final result in standard form. (Helpful fact: $2^{10} = 1024$).

Newton's Law of Universal Gravitation states that the gravitational force between two bodies is given by $F = G \frac{m_1 \cdot m_2}{d^2}$.

Knowing that the gravitational constant is $G = 6,67 \cdot 10^{-11}$ $N \cdot m^2/kg^2$, the mass of the Earth is $m_1 = 5,97 \cdot 10^{24}$ kg, the mass of the Moon is $m_2 = 7,35 \cdot 10^{22}$ kg and the mean distance between their centres is $d = 3,84 \cdot 10^8$ m, calculate the gravitational force $F$ in Newtons ($N$). Express the result in standard form rounded to two decimal places.

A glass contains exactly $18$ grams of pure water ($H_2O$). In chemistry, it is known that $18$ grams are equivalent to $1$ mol of water and that $1$ mol of any substance contains Avogadro's Number ($6,022 \cdot 10^{23}$) of molecules. From these data, calculate the mass in kilograms of a single molecule of water. Express the final result in standard form.

A research centre needs to transfer a massive backup of $4,5$ petabytes (PB) to a server in another city. The dedicated fibre-optic connection allows data to be transferred at a constant speed of $3 \cdot 10^8$ bytes per second. Knowing that $1$ PB is equivalent to $10^{15}$ bytes, calculate how many full days (and fraction of a day if necessary) it will take to complete the transfer. Show the intermediate calculations in standard form.

Two numbers $A$ and $B$ expressed strictly in scientific notation are known: $A = m \cdot 10^x$ and $B = k \cdot 10^y$, where the significands $m$ and $k$ are natural numbers ($m, k \in \{1, 2, 3, \dots, 9\}$) and the orders of magnitude $x$ and $y$ are integers ($x, y \in \mathbb{Z}$).

It is known that:

• Their product is $A \cdot B = 2,4 \cdot 10^{11}$.

• Their quotient is $\frac{A}{B} = 1,5 \cdot 10^4$.

1. a) State the relationships that the significands $m$ and $k$ and the exponents $x$ and $y$ must satisfy, taking into account whether the product $m \cdot k$ or the quotient $\frac{m}{k}$ require readjusting the power of $10$.
2. b) Analytically deduce the only pair of natural numbers $(m, k)$ that satisfies both conditions simultaneously within scientific notation. Explain why other pairs that also satisfy $\frac{m}{k} = 1,5$ are ruled out.
3. c) Calculate the integer values of $x$ and $y$, and write the numbers $A$ and $B$ explicitly.

During a torrential rain event in eastern Spain, the State Meteorological Agency (AEMET) recorded an accumulated rainfall of $180\text{ liters}$ per square meter over a region whose river basin has an area of $1,2 \cdot 10^3\text{ km}^2$.

1. a) Knowing that $1\text{ km}^2 = 10^6\text{ m}^2$ and that $1\text{ m}^3 = 10^3\text{ liters}$, calculate the total volume of water that fell on the basin in cubic meters and express it in scientific notation.
2. b) Hydrological models estimate that $60\,\%$ of that fallen water runs off into the main river channel (the rest infiltrates or evaporates). If the river has a maximum safe discharge capacity of $4,5 \cdot 10^2\text{ m}^3/\text{s}$ before overflowing, calculate how many hours it would take the channel to evacuate that volume if it flowed uninterruptedly at its maximum flow rate.
3. c) If the storm dumped that entire volume of water in an interval of just $6\text{ hours}$ and the runoff is assumed to reach the channel during those same hours, with no retention in the basin, compare this time with the result obtained in the previous part and mathematically justify why a major flood was physically inevitable, without needing to know the detailed topography of the area.