Exercises: Standard form

For a number to be correctly written in scientific notation, it must have a single non-zero digit before the decimal point.

State which of the following numbers are correctly written in scientific notation:

1. a) $12.5 \times 10^4$
2. b) $3.07 \times 10^8$
3. c) $0.95 \times 10^3$
4. d) $7 \times 10^{11}$

The blue whale is the largest animal on the planet and can weigh up to around $180\,000$ kilograms. Write this quantity in scientific notation.

Express the following astronomical distances, given in scientific notation, in ordinary decimal notation (writing all their digits):

1. a) Distance to the Moon: $3.84 \times 10^5$ km
2. b) Distance to the Sun: $1.5 \times 10^8$ km

A famous influencer has reached $2.4 \times 10^6$ followers on their main account. If they have $850\,000$ followers on their secondary account, how many followers do they have in total? Express the final result in scientific notation.

Complete in your notebook with the symbol $>$ (greater than) or $<$ (less than), comparing these numbers in scientific notation:

1. a) $4.2 \times 10^5$ ____ $1.8 \times 10^6$
2. b) $9.5 \times 10^3$ ____ $9.45 \times 10^3$
3. c) $3 \times 10^7$ ____ $2.99 \times 10^7$

A computer performs approximately $5 \times 10^9$ operations per second. How many operations will it perform in one minute ($60$ seconds)? Express the result in scientific notation.

The speed of light in a vacuum is $300\,000$ kilometres per second. If a ray of light takes $100$ seconds to reach a target, what distance has it travelled? Express the result using scientific notation.

The numbers $2$, $3$, and $4$ are plotted on an evenly spaced number line. Each unit is divided into ten equal parts by tick marks. Identify which decimal numbers correspond to the points labeled with the letters $A$ (located at the fifth tick mark between $2$ and $3$) and $B$ (located at the fifth tick mark between $3$ and $4$):

Aoife has to calculate the product $(4 \times 10^5) \times (5 \times 10^3)$ and give the final result in scientific notation. In her notebook, she has written the following steps:

When the teacher asks her how she obtained the last step, Aoife reasons: "Since $20$ is not less than $10$, I divide the mantissa by $10$ so that it becomes $2$, and to compensate, I subtract $1$ from the exponent, getting $10^7$".

1. a) Explain with reasoning what the error is in Aoife's justification about compensating the exponent.
2. b) Write the complete correction of the exercise, justifying what you really have to do with the exponent so that the numerical value is not altered.

Consider two numbers, $A$ and $B$, expressed in scientific notation: $A = 2.5 \times 10^n$ and $B = b \times 10^4$.

Knowing that their product is exactly $A \times B = 10^8$ and that the mantissa $b$ is a positive integer digit ($1 \le b < 10$):

1. a) Reason what value the mantissa $b$ must necessarily have so that, when multiplying the two mantissas ($2.5 \times b$), the result becomes an exact power of $10$.
2. b) Once that value of $b$ is fixed, calculate with reasoning the value of the integer exponent $n$, showing the steps of your deduction.

Work out the result of the following combined expression, respecting the order of operations and showing the intermediate steps:

$3^3 \times \sqrt{64} - [4^2 + (12 - 5 \times 2) \times 7]$

A famous streamer runs a subscription giveaway every $14$ days and a special Q&A live stream every $21$ days. If today, Monday, both activities happen to coincide, how many days must pass at least for both the giveaway and the special stream to occur on the same day again? What day of the week will it be?

In the school robotics club, they have $54$ infrared sensors and $72$ servomotors. They want to organise the equipment into identical boxes with the largest possible number of components in each, without mixing types and with nothing left over.

1. a) How many components will be in each box?
2. b) How many boxes of sensors and how many boxes of servomotors will they get?

Simplify to a single power using the laws of indices:

1. a) $(2^5 \times 2^3)^2 : 2^{10}$
2. b) $\frac{(3^4)^3 \times 9^2}{27^3}$ (Hint: first express all numbers as powers of base 3)

The average distance from the Earth to the Sun is approximately $149\,600\,000$ kilometres.

1. a) Express this quantity in metres using scientific notation.
2. b) State the order of magnitude of this distance expressed in metres.

Write the following scientific data, taken from a technology magazine, in ordinary decimal form (extended form):

1. a) The number of transistors in an advanced microprocessor is $4.5 \times 10^9$.
2. b) The transfer speed of a fibre optic cable is $2.4 \times 10^7$ bits per second.
3. c) The storage capacity of a server is $1.25 \times 10^{12}$ bytes.

A red blood cell has a diameter of approximately $7.5 \times 10^{-6}$ metres. If we could line up $100$ million red blood cells side by side, what total length would they reach? Express the final result in scientific notation and in kilometres.

Consider the number $N = 3 \times 10^a$. Knowing that the square of this number is $90\,000$:

1. a) Find the value of the exponent $a$.
2. b) Write the number $N$ in ordinary decimal notation.
3. c) Is $N$ a perfect square? Justify your answer by calculating its square root.

According to data published by the Central Statistics Office (CSO), the resident population in Ireland is approximately $4.8 \times 10^7$ people. In a full year, Irish households and municipalities as a whole generate around $2.4 \times 10^7$ metric tonnes of municipal waste.

1. a) Knowing that $1\text{ tonne} = 10^3\text{ kg}$, write the total amount of waste generated per year in kilograms in scientific notation.
2. b) Calculate the average amount of waste generated by an Irish resident in a year (in $\text{kg/inhabitant}$). Express the operation and the result in scientific notation.
3. c) Taking the year to have $365$ days, calculate how many grams of rubbish an average person in Ireland produces daily. Write the final result in standard decimal notation and interpret whether the obtained result is consistent with reality.

An environmental research team collects a sediment sample in the Wexford Slobs wetlands on the Wexford coast. Microscopic analysis determines that there is an average concentration of $3.8 \times 10^8$ microorganisms per gram of soil. The total mass of the collected sample is $4.2 \times 10^2\text{ grams}$.

1. a) Estimate: Without using a calculator or doing detailed operations on paper, round the mantissas to the nearest units and deduce mentally the order of magnitude (the exponent of the power of $10$) that the total number of microorganisms in the sample will have. Briefly explain how you reasoned it.
2. b) Calculate: Perform the multiplication with all its steps applying the properties of scientific notation and express the result with its proper mantissa.
3. c) Check: Compare your initial estimate with the exact result obtained. Did the order of magnitude match? Justify why making a prior estimate helps detect calculation errors.

There are approximately $2.5 \times 10^{13}$ red blood cells in the body of a teenager. If each of these cells has a diameter of just $7 \times 10^{-6}$ metres, imagine you could arrange them all end to end in a single line, forming a very long chain.

Calculate the total length of this chain in kilometres and express the result in standard form. How many times could this chain wrap around the Earth if the circumference of the equator is approximately $4 \times 10^4$ km?

The mean distance from the Earth to the Sun is $1.496 \times 10^8$ km, while the mean distance from Mars to the Sun is $2.279 \times 10^8$ km. At a given moment, the three celestial bodies are perfectly aligned, with the Earth located between the Sun and Mars.

Work out the distance between the Earth and Mars at that instant. Express the result in standard form and in metres.

A server for a popular video-sharing social network receives an average of $4.5 \times 10^5$ megabytes (MB) of data every second.

1. a) How many gigabytes (GB) of data does the server receive in one hour? (Remember that $1$ GB = $10^3$ MB).
2. b) If the server has a total storage capacity of $1.2 \times 10^9$ GB, how many days would it take to fill up completely if no data were deleted? Round your final answer to the nearest whole number.

In chemistry, it is known that one mole of any substance contains $6.022 \times 10^{23}$ molecules (Avogadro's number). If we have a tank containing $2.5 \times 10^2$ moles of water and we know that the mass of a single molecule of water is approximately $3 \times 10^{-23}$ grams:

1. a) Calculate the total number of water molecules in the tank.
2. b) Calculate the total mass of the water in the tank in kilograms. Express both answers in standard form.

Light travels at a constant speed of $3 \times 10^8$ metres per second. A latest-generation processor performs a basic operation every $0.5$ nanoseconds (one nanosecond is $10^{-9}$ seconds).

What distance does light travel in the exact time it takes the processor to perform a single operation? Express the result in millimetres and in standard form.

A power plant produces $2.8 \times 10^{10}$ kilowatt-hours (kWh) annually. In a city, there are $5 \times 10^5$ households, and each household consumes an average of $3.5 \times 10^3$ kWh per year.

Calculate how much surplus energy is left over after supplying the whole city. Express the result in standard form and consider: could this surplus energy supply another city of $2 \times 10^5$ households with the same average consumption? Justify your answer by showing the necessary calculations.

If all the DNA from a single human cell were uncoiled, it would measure approximately $2 \times 10^0$ metres (that is, 2 metres). The human body contains around $3.7 \times 10^{13}$ cells.

1. a) What would be the total length of all the DNA in a human body if we could join it all together? Express it in kilometres and standard form.
2. b) Knowing that the distance from the Earth to the Moon is $3.84 \times 10^5$ km, how many round trips to the Moon could be made with all that DNA?

Order the following values from least to greatest, but before doing so, express all of them as a "normal" decimal number (standard notation) to check your accuracy:

• $A = 345 \times 10^6$

• $B = 0.034 \times 10^{10}$

• $C = 3.5 \times 10^8$

• $D = 3.400 \times 10^5$

Once ordered, perform the following operation and express the final result in scientific notation: $\frac{A + C}{D}$. (Note: Be very careful with the exponents when adding).

In a 2nd Year class, students are asked to calculate the result of the following subtraction: $R = 3.2 \times 10^{14} - 7.5 \times 10^{11}$. Two students defend different strategies:

• Seán says: "I prefer to write both numbers with all their digits and zeros in the traditional decimal system, place one below the other, subtract, and then convert the result back to scientific notation".

• Aoife replies: "That is very risky. It is much better to make the exponents equal in scientific notation by transforming one of the terms, so we can factor out the power of $10$ and subtract only the mantissas".

1. a) Analyse the drawbacks and calculation risks of Seán's method in this specific case, stating how many zeros he would have to write for each number.
2. b) Apply Aoife's method step by step: equalise the exponents to the larger one ($10^{14}$), perform the subtraction, and give the solution in correct scientific notation.
3. c) Describe in which particular situation of an addition or subtraction it would be reasonable to apply Seán's method, and give a numerical example with powers of $10$ that justifies it.

In an astronomical observatory, two celestial bodies, $X$ and $Y$, are analysed, whose estimated masses are given by $M_X = 4.8 \times 10^k\text{ kg}$ and $M_Y = 1.6 \times 10^{k-2}\text{ kg}$, where $k$ is a positive integer known to be greater than $5$.

1. a) Without knowing the specific numerical value of $k$, calculate how many times more massive body $X$ is than body $Y$. Justify in detail why the result obtained is independent of the value of $k$.
2. b) If both bodies were to collide and merge into a single object of mass $M_T = M_X + M_Y$, calculate that combined mass expressed in scientific notation in terms of $k$ (with its mantissa properly between $1$ and $10$).
3. c) If the astronomers state that the combined mass $M_T$ is strictly less than $10^{19}\text{ kg}$, find the largest possible integer value that the exponent $k$ can take. Reason your deduction using an inequality.