Scientific notation

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.

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 $4$):

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

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.

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.

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) $3 \times 10^7$ ____ $2.99 \times 10^7$

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).

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.

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.

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.

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)

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.