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 $300\,000$ kilometers per second. If a ray of light takes $100$ seconds to reach a target, what distance has it traveled? 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 $4$):

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 $149\,600\,000$ kilometers.

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

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 fiber 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 $7.5 \times 10^{-6}$ meters. 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 kilometers.

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.

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 $7 \times 10^{-6}$ meters, 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 kilometers and express the result in scientific notation. 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 meters.

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 $3 \times 10^8$ 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 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 $2 \times 10^0$ meters (that is, 2 meters). 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 kilometers and scientific notation.
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).