Exercises: Standard form

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) $(10^2)^3 \times 10^4$

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) $4^2 \times 3^2 = 12^2$

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 scientific notation to measure things that we cannot see with the naked eye.

Write the following measurements in scientific notation:

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

Look at the image to visualize 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) $8.9 \times 10^{-5}$

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.

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.

An influencer is analysing their social media statistics to see the growth of their audience.

• On Instagram they have $1.25 \times 10^5$ followers.

• On TikTok they have $8.4 \times 10^4$ followers.

• On YouTube they have $2.1 \times 10^5$ subscribers.

1. a) Calculate the total number of followers by adding the three platforms. Express the result in standard form.
2. b) If they expect their total audience to grow by $10\%$ next month, how many followers will they have in total?

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 meters using scientific notation.
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 $e = v \cdot t$).

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?

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?

A cubic reservoir has an edge of $4 \cdot 10^2$ meters.

1. a) Calculate the volume of the reservoir in cubic meters ($m^3$) and express the result in scientific notation.
2. b) Given that $1 \text{ m}^3 = 1000 \text{ liters}$, calculate the total capacity of the reservoir in liters, also expressed in scientific notation.
3. c) If the reservoir is only $50\%$ full, how many liters of water does it contain?

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) $\frac{2^4 \times (2^3)^2}{2^{-2}}$

A water molecule ($H_2O$) is formed by two hydrogen atoms and one oxygen atom. The masses of these atoms are approximately:

• Mass of Hydrogen atom: $1.67 \times 10^{-24}$ g

• Mass of Oxygen atom: $2.66 \times 10^{-23}$ g

1. a) In order to add the masses, express the mass of the oxygen atom as a number multiplied by $10^{-24}$.
2. b) Calculate the total mass of a single water molecule ($2 \times \text{Hidrógeno} + 1 \times \text{Oxígeno}$). Express the final result in standard form.

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.

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

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

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

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

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

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

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.

Work out the following combined operations in standard form without using a calculator for intermediate steps. You must adjust the exponents to add or subtract and apply the properties to multiply or divide. Express the final result in correctly normalised standard form.

$$ \frac{(3.2 \times 10^{-4}) + (1.8 \times 10^{-3})}{2 \times 10^{-2}} \times (4 \times 10^6) $$

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.).

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).

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

A sheet of paper has an approximate thickness of $10^{-4}$ meters. Imagine that we could fold the sheet in half as many times as we wanted.

1. a) Write an expression in power form that gives the thickness of the stack of paper after making $n$ folds.
2. b) If we managed to fold the sheet 42 times (physically impossible, but mathematically calculable), what would the thickness of the stack be in kilometers?
3. c) Compare the result with the mean distance from the Earth to the Moon ($3.84 \cdot 10^5$ km). Would the thickness of the paper exceed this distance?