Scientific notation

In our daily lives, and especially in technology, we use very large numbers. Write the following quantities related to social media and the internet in standard form.

Remember that standard form has the form $a \cdot 10^n$, where $1 \le a < 10$.

1. a) Number of views of the most viewed video of the month: 12 500 000 views.
2. b) Number of bytes in a Terabyte (TB): 1 000 000 000 000 bytes.
3. c) Approximate number of users of a social network: 2 900 000 000 users.
4. d) Your age in seconds (approximately, calculating 14 years): 441 500 000 seconds.

Now let us work with very small things. Biology and physics use standard form to measure things that we cannot see with the naked eye.

Write the following measurements in standard form:

1. a) The diameter of a red blood cell: 0.0000075 metres.
2. b) The approximate size of a flu virus: 0.00000012 metres.
3. c) The mass of a fine grain of dust: 0.00000067 grams.
4. d) The time it takes light to travel one metre: 0.0000000033 seconds.

Look at the image to visualise the microscopic scale:

Convert the following numbers given in standard form into decimal form (standard number).

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

In the study of natural sciences we come across extremely large and extremely small lengths. Four real lengths are given below, all expressed in meters:

• Diameter of the flu virus: $0,00000012$ m

• Thickness of a human hair: $0,00007$ m

• Diameter of the Earth: $12742000$ m

• Distance from the Earth to the Sun: $149600000000$ m

Your task is to:

1. a) Write each of the lengths in scientific notation.

1. b) Order them from smallest to largest.

Order the following numbers from smallest to largest. To do this, look closely at the exponent of the power of 10. If the exponents are equal, compare the decimal numbers.

$$ 3.2 \cdot 10^5 \quad ; \quad 1.5 \cdot 10^6 \quad ; \quad 8.9 \cdot 10^4 \quad ; \quad 3.1 \cdot 10^5 $$

Use the symbol $<$ to write the final answer.

The speed of light in a vacuum is approximately $3 \cdot 10^8$ m/s. A spacecraft sends a radio signal from Mars to Earth. If the distance between the two planets at that moment is $2,25 \cdot 10^8$ km:

1. a) Express the distance in metres using standard form.
2. b) Calculate how long (in seconds) it takes the signal to reach Earth.
3. c) Express that time in minutes.

(Remember that speed $v$, distance $e$ and time $t$ are related by $e = v \cdot t$).

The human body contains approximately $3,7 \cdot 10^{13}$ cells. It is estimated that each cell has an average mass of $2 \cdot 10^{-9}$ grams.

1. a) Calculate the total mass of a person's cells in grams.
2. b) Convert the previous result to kilograms.
3. c) If a bacterium weighs $10$ times less than a human cell, what is the mass of a bacterium in standard form?

Simplify the following expressions using the properties of powers and express the final result as a single power with a positive exponent.

1. a) $\frac{3^5 \cdot 3^{-2}}{3^7}$
2. b) $(5^3 \cdot 5^{-4})^{-2}$
3. c) $\frac{2^4 \cdot (2^3)^2}{2^{-2}}$