Scientific notation

Calculate the value of the following powers. Pay close attention to the use of brackets and the sign of the base, as this is one of the most common mistakes in this topic.

1. a) $2^4$
2. b) $(-2)^4$
3. c) $-2^4$
4. d) $(-3)^3$
5. e) $-3^3$
6. f) $5^0$

Look at the following image, which geometrically represents the difference between $2^2$ and $2^3$. Could you draw in your notebook how $3^2$ would be represented?

Use the properties of powers to simplify the following expressions and write them as a single power. You do not need to calculate the final result, only express the resulting power.

Remember that:

• To multiply powers with the same base, we add the exponents.

• To divide powers with the same base, we subtract the exponents.

• To raise a power to another power, we multiply the exponents.

1. a) $3^5 \times 3^2$
2. b) $7^8 : 7^3$
3. c) $(2^3)^4$
4. d) $x^4 \times x \times x^2$
5. e) $\frac{5^7}{5^2}$
6. f) $(10^2)^3 \times 10^4$

We have a giant cubic tank designed to store water for irrigation. The edge of this tank measures $2^5$ metres.

1. a) Express the volume of the tank in cubic metres using a single power with base 2.
2. b) Given that $1 \text{ m}^3 = 1000 \text{ litres}$, express the total capacity of the tank in litres using standard form.
3. c) If the tank is half full, how many litres does it contain? Express the result as a power with base 2 multiplied by a power with base 10.

Convert the following powers with negative exponents into fractions with positive exponents, and vice versa. Calculate the final value where possible.

This exercise will help you become fluent in using the property $a^{-n} = \frac{1}{a^n}$.

1. a) $2^{-3}$
2. b) $5^{-2}$
3. c) $\frac{1}{3^2}$
4. d) $10^{-1}$
5. e) $\frac{1}{2^{-4}}$
6. f) $(\frac{2}{3})^{-2}$

State whether the following statements are True or False. If any are false, write the correct expression in your notebook.

1. a) $2^3 \times 2^4 = 2^{12}$
2. b) $3^{-2} = -9$
3. c) $(5^2)^3 = 5^6$
4. d) $\frac{a^5}{a^5} = 0$
5. e) $4^2 \times 3^2 = 12^2$

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

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

Simplify the following algebraic expression using the properties of powers. You must break down the bases into prime factors, calculate, and express the final result as an irreducible fraction with positive exponents. It is not necessary to calculate the final numerical value, only the simplified power.

$$ \frac{15^{-2} \times 4^3 \times (-3)^4}{(-10)^2 \times (2^{-2})^{-3} \times 9^{-1}} $$

Find the value of $x$ in the following exponential equation using the properties of powers. To do this, express all the numbers as powers of the same base (base 3).

$$ \frac{3^x \times 9^2}{27^{-1}} = \left( \frac{1}{3} \right)^{-5} $$

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 $a \times 10^n$, 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) $3.5 \times 10^4$
2. b) $1.2 \times 10^6$
3. c) $5.4 \times 10^{-3}$
4. d) $8.9 \times 10^{-5}$

To express the number $0.0045 \times 10^5$ in scientific notation, Aoife reasons as follows in her notebook:

"To have a single non-zero digit before the decimal point, I move the decimal point three places to the right. Since I moved the decimal point three places to the right, I add 3 to the exponent of the power of ten. Therefore:

$$0.0045 \times 10^5 = 4.5 \times 10^{5+3} = 4.5 \times 10^8$$"

1. a) Explain in detail why the initial expression $0.0045 \times 10^5$ does not satisfy the definition of scientific notation.
2. b) Identify the error in Aoife's reasoning when modifying the exponent and explain how shifting the decimal point is actually compensated so that the value of the number does not change.
3. c) Write the correct step-by-step procedure and give the final result both in scientific notation and as an ordinary decimal number.

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.

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

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

The speed of light in a vacuum is approximately $3 \times 10^8$ m/s. A spacecraft sends a radio signal from Mars to Earth. If the distance between the two planets at that moment is $2.25 \times 10^8$ 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 $e = v \times t$).

The human body contains approximately $3.7 \times 10^{13}$ cells. It is estimated that each cell has an average mass of $2 \times 10^{-9}$ 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?

We want to obtain the number $3.6 \times 10^{-2}$ as an exact result.

1. a) Come up with two numbers, $A$ and $B$, both correctly expressed in scientific notation, such that the order of magnitude of $A$ is $-5$ and $A \times B = 3.6 \times 10^{-2}$ holds true.
2. b) Show how you deduced the mantissa and the order of magnitude of $B$ from the value of $A$ you chose.
3. c) Would it be possible to choose a number $A$ with an order of magnitude of $-5$ such that the order of magnitude of $B$ had to be exactly $2$? Justify your answer taking into account the possible values that the product of two valid mantissas can take (remember that each mantissa must lie in the interval $[1, 10)$).

According to data from the Central Statistics Office (CSO), households in a certain region consumed a total of $2.38 \times 10^8\text{ m}^3$ of drinking water in one year, distributed among a population of $4.75 \times 10^6$ inhabitants.

1. a) Estimate: Without doing operations on paper or using a calculator, round the mantissas to simple whole numbers and estimate mentally how many cubic meters of water per year correspond to each inhabitant. State the expected order of magnitude for the result.
2. b) Calculate: Perform the division using scientific notation with the original data. Express the quotient in scientific notation with the mantissa rounded to the hundredths place.
3. c) Check and assess: Knowing that $1\text{ m}^3 = 10^3\text{ liters}$ and considering a year of 365 days, calculate how many liters of water per day an inhabitant consumes on average according to that data. Compare the result with your previous estimate and reason whether you find it a plausible figure for the daily consumption of a person in a household.

The Voyager 1 space probe is the furthest man-made object from Earth. Imagine that it is at an approximate distance of $2.3 \times 10^{10}$ kilometres from Earth.

If the probe sends a radio signal towards us, and knowing that signals travel at the speed of light ($3 \times 10^5$ km/s):

1. a) Calculate how long (in seconds) it takes the signal to reach Earth. Express the result in standard form.
2. b) Convert that time to hours and express it as a decimal number rounded to two decimal places.

The average adult human body has approximately $3.72 \times 10^{13}$ cells. Assuming that, on average, a human cell has a mass of $1.8 \times 10^{-9}$ grams:

1. a) Calculate the total mass of all the cells in grams.
2. b) Convert the result to kilograms.
3. c) If a person weighs 75 kg, approximately what percentage of their total weight corresponds to the calculated cell mass? (Note: the rest would be extracellular water, bone minerals, etc.).

A cloud storage service offers its users 2 TB (Terabytes) of space. Given that the approximate equivalences in the computer system are:

• $1 \text{ TB} = 10^{12} \text{ bytes}$

• $1 \text{ GB} = 10^9 \text{ bytes}$

• $1 \text{ MB} = 10^6 \text{ bytes}$

If a user has a photo collection where each photo takes an average of 4 MB:

1. a) How many total bytes of storage does the user have available? Express it in standard form.
2. b) Approximately how many photos can they store in their cloud before it is full?

In the world of computing, storage units grow rapidly. A data server has a total storage capacity of 4 Petabytes.

Given that:

• $1 \text{ Petabyte (PB)} = 10^{15} \text{ bytes}$

• A high-quality photograph takes an average of $8 \text{ Megabytes (MB)}$, where $1 \text{ MB} = 10^6 \text{ bytes}$.

Calculate how many high-quality photographs fit on this server. Express the final result in standard form.

Challenge of the unit. A sheet of paper has a thickness of approximately $10^{-4}$ meters. Imagine we could fold the sheet in half as many times as we wanted.

1. a) Write an expression in the form of a power that gives the thickness of the stack of paper after $n$ folds.

1. b) If we managed to fold the sheet 42 times (something physically impossible, but mathematically computable), what would the thickness of the stack be in kilometers? Express the result in scientific notation.

1. c) Compare the result with the average distance from the Earth to the Moon ($3.84 \times 10^5$ km). Would the thickness of the stack exceed this distance?