Exercises: Standard form

For a number to be correctly expressed in standard form, it must have a single non-zero digit before the decimal point.

State which of the following numbers are written correctly in standard form:

1. a) $12,5 \cdot 10^4$
2. b) $3,07 \cdot 10^8$
3. c) $0,95 \cdot 10^3$
4. d) $7 \cdot 10^{11}$

The blue whale is the largest animal on the planet and can weigh up to about $180\,000$ kilograms. Write this amount in standard form.

Express in decimal notation (writing all its digits) the following astronomical distances given in standard form:

1. a) Distance to the Moon: $3,84 \cdot 10^5$ km
2. b) Distance to the Sun: $1,5 \cdot 10^8$ km

A famous influencer has reached the figure of $2,4 \cdot 10^6$ 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 standard form.

Complete in your exercise book with the sign $>$ (greater than) or $<$ (less than) comparing these numbers in standard form:

1. a) $4,2 \cdot 10^5$ ____ $1,8 \cdot 10^6$
2. b) $9,5 \cdot 10^3$ ____ $9,45 \cdot 10^3$
3. c) $3 \cdot 10^7$ ____ $2,99 \cdot 10^7$

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

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

Look at the following number line where the powers of base $10$ are represented. Identify which decimal numbers correspond to the points marked with the letters $A$ and $B$:

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

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

A famous streamer holds a subscription giveaway every $14$ days and a special Q&A live stream every $21$ days. If today, Monday, they happen to have done both activities, what is the minimum number of days that must pass for them to do the giveaway and the special live stream 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 organize the equipment into identical boxes with the greatest possible number of components, without mixing types and with nothing left over.

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

Reduce to a single power using the properties of powers:

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

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

1. a) Express this amount in meters using standard form.
2. b) State the order of magnitude of this distance expressed in meters.

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

1. a) The number of transistors in an advanced microprocessor is $4,5 \cdot 10^9$.
2. b) The transfer speed of a fiber optic cable is $2,4 \cdot 10^7$ bits per second.
3. c) The storage capacity of a server is $1,25 \cdot 10^{12}$ bytes.

A red blood cell has an approximate diameter of $7,5 \cdot 10^{-6}$ meters. If we could place $100$ million red blood cells in a row next to each other, what total length would they reach? Express the final result in standard form and in kilometers.

Consider the number $N = 3 \cdot 10^a$. If we know 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 decimal notation.
3. c) Is $N$ a perfect square? Justify your answer by calculating its square root.

In a teenager's body there are approximately $2,5 \cdot 10^{13}$ red blood cells. If each of these cells has a diameter of just $7 \cdot 10^{-6}$ meters, imagine you could place them all in a row, one after the other, forming a very long chain.

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

The average distance from the Earth to the Sun is $1,496 \cdot 10^8$ km, while the average distance from Mars to the Sun is $2,279 \cdot 10^8$ km. At a given moment, the three celestial bodies are perfectly aligned, with the Earth located between the Sun and Mars.

Calculate the distance separating the Earth from Mars at that moment. Express the result in standard form and in meters.

A server for a popular video social network receives an average of $4,5 \cdot 10^5$ megabytes (MB) of information every second.

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

In chemistry, it is known that one mole of any substance contains $6,022 \cdot 10^{23}$ molecules (Avogadro's Number). If we have a tank containing $2,5 \cdot 10^2$ moles of water and we know that the mass of a single water molecule is approximately $3 \cdot 10^{-23}$ 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 results in standard form.

Light travels at a constant speed of $3 \cdot 10^8$ meters 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 millimeters and in standard form.

A power station produces $2,8 \cdot 10^{10}$ kilowatt-hours (kWh) annually. In a city, there are $5 \cdot 10^5$ homes, and each home consumes an average of $3,5 \cdot 10^3$ kWh per year.

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

If we stretched out all the DNA from a single human cell, it would measure approximately $2 \cdot 10^0$ meters (that is, 2 meters). The human body has around $3,7 \cdot 10^{13}$ cells.

1. a) What would be the total length of all the DNA in a person if we could join it together? Express it in kilometers and standard form.
2. b) Knowing that the distance from the Earth to the Moon is $3,84 \cdot 10^5$ km, how many times could we go to the Moon and back with all that DNA?

Order the following values from smallest to largest, but before doing so, express them all as a "normal" decimal number (ordinary notation) to compare your precision:

• $A = 345 \cdot 10^6$

• $B = 0,034 \cdot 10^{10}$

• $C = 3,5 \cdot 10^8$

• $D = 3.400 \cdot 10^5$

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