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 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) $8.9 \times 10^{-5}$

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.

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.

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 $[1, 10)$).

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 $3 \times 10^8$ 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 $e = v \times t$).

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 $4 \times 10^2$ metres.

1. a) Calculate the volume of the reservoir in cubic metres ($m^3$) and express the result in standard form.
2. b) Given that $1 \text{ m}^3 = 1000 \text{ litres}$, calculate the total capacity of the reservoir in litres, also expressed in standard form.
3. c) If the reservoir is only $50\%$ full, how many litres 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 ($2 \times \text{Hydrogen} + 1 \times \text{Oxygen}$). Express the final result in standard form.

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.

During a torrential rain event monitored by Met Éireann on the west coast of Ireland, a storm dropped an average of $1.2 \times 10^2\text{ l/m}^2$ over a region of $3.5 \times 10^2\text{ km}^2$ in area. It is estimated that a storm raindrop has a volume of approximately $4 \times 10^{-5}\text{ liters}$.

1. a) Express the area of the region in square meters ($\text{m}^2$) in scientific notation, knowing that $1\text{ km}^2 = 10^6\text{ m}^2$.
2. b) Model the total precipitation by calculating the overall volume of water (in liters) that fell on the region during the event. Express the result in scientific notation.
3. c) Calculate the total number of raindrops that fell over the entire region.
4. d) Observe the order of magnitude of the number of drops obtained and write a brief reflection arguing why scientific notation is essential in meteorological and environmental studies compared to decimal notation with zeros.

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 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.

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.

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).

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

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?

In an interdisciplinary Science project, the aim is to calculate the total mass of the Earth's hydrosphere and atmosphere by adding their estimated masses: the mass of the atmosphere is $M_A = 5.15 \times 10^{18}\text{ kg}$ and that of the hydrosphere is $M_H = 1.4 \times 10^{21}\text{ kg}$. To calculate $M_T = M_A + M_H$, three classmates propose different strategies:

• Method 1 (Cian): Express the larger quantity in the smaller power ($1.4 \times 10^{21} = 1400 \times 10^{18}$), add the mantissas, and finally adjust to scientific notation.

• Method 2 (Aoife): Express the smaller quantity in the larger power ($5.15 \times 10^{18} = 0.00515 \times 10^{21}$), add the mantissas, and obtain the result directly in scientific notation.

• Method 3 (Seán): Write both numbers with all their zeros in standard decimal notation, align them vertically to add digit by digit, and convert the final result back to scientific notation.

1. a) Solve the addition by explicitly applying Method 1 and Method 2, verifying that both paths lead to the same result.
2. b) Compare the three methods: justify which one is the most efficient and reliable in this case, and argue why Seán's approach is completely inadvisable when working with scientific quantities.
3. c) Analyze the result obtained: what approximate percentage does the mass of the atmosphere represent relative to that of the hydrosphere? Explain how this disproportion influences the order of magnitude of the final sum.

The deep space tracking station located in the Burren (County Clare) receives data from a space probe. The distance between the probe and Earth varies between a perihelion of $4.5 \times 10^7\text{ km}$ and an aphelion of $1.5 \times 10^8\text{ km}$. Electromagnetic signals travel at the speed of light, $c = 3 \times 10^5\text{ km/s}$. In addition, the probe sends a data packet of $7.2 \times 10^6\text{ kilobits}$ at a constant transmission rate of $2.4 \times 10^4\text{ kilobits/s}$. The "total reception time" is the sum of the time the signal takes to travel through space plus the time the antenna takes to download the entire packet.

1. a) Set up the formula for the total reception time in terms of the given data and calculate this time (in seconds and in scientific notation) at perihelion and at aphelion.
2. b) Due to operational constraints of the control center, an emergency protocol is established requiring that the total reception time of said packet must not under any circumstances exceed $400\text{ seconds}$. Set up an inequality and calculate the maximum distance to Earth (in km and in scientific notation) at which the spacecraft can be to meet that requirement.
3. c) Interpret the result obtained in part b): will the spacecraft be within the safety limit during its entire orbit, nowhere in it, or only in a specific section? Justify your answer by comparing the maximum distance found with the perihelion and the aphelion.

Topic challenge. Design a scaled energy model for your town using real data from EirGrid.

1. Data collection and scientific notation: Access the EirGrid website (system demand section) and look up yesterday's total all-island consumption in $\text{MWh}$. Convert it to watt-hours ($\text{Wh}$) by multiplying by $10^6$ and express the final value, rounded to two significant figures, in scientific notation ($a \times 10^k$, with $1 \le a < 10$).

2. Powers of rational numbers: Assume that your municipality accounts for a fraction equivalent to $(2/5)^3$ of one thousandth of that total energy. Use the properties of powers with a rational base to simplify the operation and find the estimated consumption of your town in scientific notation.

3. Operations with radicals: If a standard solar panel yields $2.5 \times 10^2\text{ Wh}$ daily per square meter, calculate the total area $A$ of panels needed. Plan the installation on a square plot of land with side length $L = \sqrt{A}$ meters. Factor the radicand, take all possible factors out of the root to obtain the simplified exact solution, and finally, approximate its value with a decimal point.

Final product: Submit a brief one-page report (or a screenshot of your spreadsheet) with the collected data, detailed mathematical workings, and the interpretation of the result.

Assessment criteria:

• Precision in unit conversion and rigorous use of scientific notation.

• Correct application of the properties of powers of rational numbers.

• Prime factorization and skill in simplifying and taking factors out of radicals.

• Clarity, order, and justification of conclusions in the report.