Exercises: Laws of indices

In the school games room, board games are organised on shelves. If we have an expression in power form such as $7^{3}$, state which number is the base and which is the exponent. Briefly explain what operation you would need to perform to calculate its value (without needing to give the final result).

A well-known social network displays user profile photos in a square grid. If Lucía's photo feed is $5$ photos wide and $5$ photos high, how many photos does she have in total? Express the result first as a multiplication of two equal factors, then as a power, and finally calculate the total value.

Calculate the following special powers mentally. Remember the rules for exponents $0$ and $1$:

1. a) $15^{1} =$
2. b) $8^{0} =$
3. c) $1^{25} =$
4. d) $10^{1} =$
5. e) $2024^{0} =$

Apply the rule for the product of powers with the same base to write the following expressions as a single power (it is not necessary to calculate the final value):

1. a) $3^{2} \cdot 3^{4} =$
2. b) $5^{3} \cdot 5^{3} =$
3. c) $a^{2} \cdot a^{5} =$
4. d) $10^{2} \cdot 10 \cdot 10^{3} =$ (Hint: when a number has no exponent, it is as if it has a $1$).

Use the rule for the quotient of powers with the same base to simplify these operations into a single power:

1. a) $7^{5} : 7^{2} =$
2. b) $10^{8} : 10^{6} =$
3. c) $\frac{4^{9}}{4^{7}} =$
4. d) $m^{10} : m^{9} =$

Write as a single power using the power of a power rule:

1. a) $(2^{3})^{2} =$
2. b) $(5^{4})^{3} =$
3. c) $(10^{2})^{5} =$
4. d) $(x^{3})^{4} =$

In a video game, each time you go up a level, your points are multiplied by $10$. If you have passed $4$ levels and the score is written as $10^{4}$, how many points do you have in total? Write the number with all its digits and explain how many zeros you have added.

Calculate the following exact square roots by thinking about which number multiplied by itself gives the radicand. Look at the example in the figure: $\sqrt{25} = 5$ because $5^{2} = 25$.

1. a) $\sqrt{16} =$

1. b) $\sqrt{49} =$

1. c) $\sqrt{81} =$

1. d) $\sqrt{100} =$

In a Maths test, Lucía has to simplify the operation $3^4 \cdot 3^2$ and leave it as a single power. On her paper she has written the following: "$3^4 \cdot 3^2 = 9^6$, because I multiply the bases ($3 \cdot 3 = 9$) and add the exponents ($4 + 2 = 6$)".

1. a) Identify the mistake made by Lucía and which part of her reasoning is correct.
2. b) Explain in your own words, developing the meaning of powers as repeated factors, why the base must be kept when multiplying powers with the same base.
3. c) Write the correct solution as a single power.

You are asked to mentally calculate the value of the expression $2^4 \cdot 5^4$.

1. a) Before doing any calculations on paper, make an estimate: will the final result have three, four, or five digits? Briefly explain what you base your prediction on.
2. b) Apply the power of a product property in reverse (that is, using $a^n \cdot b^n = (a \cdot b)^n$) to calculate the exact value of the expression immediately.
3. c) Check whether your estimate was correct and explain how the common exponent is related to the number of zeros in the final result.

Use the laws of indices to simplify the following expression and leave it as a single power with base $2$:

Remember to show the steps followed: first work out the inside of the brackets, then apply power of a power, and finally work out the quotient.

In the computing world, storage capacities are usually expressed as powers with base $10$. An external hard drive has a capacity of $10^3$ gigabytes ($GB$). If each gigabyte contains $10^3$ megabytes ($MB$), and each megabyte has $10^6$ bytes:

1. a) Express the total capacity of the hard drive in bytes using a single power of $10$.
2. b) How many photos of $10^6$ bytes each would fit on the hard drive? Express the result as a power.

A courier company organises its packages in large cubic containers. If a container fits $6$ boxes along its length, $6$ boxes along its width and $6$ boxes along its height:

1. a) Express the total number of boxes in a container as a power and calculate the result.
2. b) If the company has $6^2$ identical containers stored, how many boxes do they have in total? Express the result as a single power.

Calculate the following expressions in two different ways to check that the laws of indices hold true:

• Method A: By calculating the inside of the brackets first and then the power.

• Method B: By applying the rule for the power of a product or a quotient.

1. a) $(2 \cdot 5)^3$
2. b) $(12 : 3)^2$
3. c) $(2^2 \cdot 3)^2$

A school charity challenge is spread by chain reaction. At first, $3$ students propose it. Each of them proposes it to $3$ other new people, and so on across different levels of sharing.

1. a) How many people will have received the challenge at the fifth level? Express it as a power and calculate the value.
2. b) If at level $n$, $243$ people have received it, what is the value of $n$?
3. c) Express the product of the third level and the fourth level as a single power.

A school playground is square-shaped and has a total area of $225 m^2$. The maintenance team wants to surround it with barrier tape for a sports activity, leaving a gap of $2$ metres for a gate on one side.

1. a) How long is the side of the playground? (Use the square root).
2. b) How many metres of tape do they need to buy?

Solve the following mixed operations, applying the order of operations correctly and the laws of indices where possible:

1. a) $3^3 - \sqrt{49} + 2^2 \cdot (5 - 3)^2$
2. b) $(\sqrt{100} - 2^3)^2 + 4^0 \cdot 5$
3. c) $2^5 : 2^3 + \sqrt{81} \cdot 3 - 1^{15}$

Find the value of the letter $x$ in each of the following equations using the laws of indices:

1. a) $5^x \cdot 5^4 = 5^9$
2. b) $(3^2)^x = 3^6$
3. c) $10^7 : 10^x = 10^3$
4. d) $x^3 = 64$
5. e) $x^0 = 1$ (State which values $x$ cannot take in this case).

In a regional school 3x3 basketball tournament organised in Zaragoza, $2^6$ teams participate. The championship is played by single-elimination: in each round the teams play in pairs, so that only the winners advance to the next phase (in each round the number of participating teams is divided by $2$).

1. a) Express, using a division of powers with the same base, how many teams will remain in the competition after $4$ elimination rounds have been played. Apply the properties of powers to simplify the expression to a single power of base $2$ and calculate how many teams remain.
2. b) To crown the champion team, exactly $1$ team must remain. Use the quotient of powers with the same base property and the definition of a zero exponent ($2^0 = 1$) to deduce how many elimination rounds will be played in total until the end of the tournament.

To simplify the expression $(18^3 : 6^3) \cdot 3^5$ and leave it expressed as a single power, two students propose different approaches:

• Method 1 (Marcos): Observes that the parentheses contain a division with the same exponent ($3$), so he first calculates $(18 : 6)^3$ and then multiplies the result by $3^5$.

• Method 2 (Elena): Prefers to first decompose the bases into prime factors ($18 = 2 \cdot 3^2$ and $6 = 2 \cdot 3$), applies the properties of powers to all factors, and finally simplifies.

1. a) Apply Marcos's Method 1 step by step, stating in each line which property of powers you are using, and find the result as a single power.
2. b) Explain, with reasoning, why Marcos's method is faster and reduces the likelihood of making calculation errors in this case.
3. c) Give an example of an operation where Marcos's method cannot be directly applied and Elena's prime factorisation would be necessary (for instance, if the bases were not divisible).

Apply the laws of indices and perform the necessary base changes to express the result as a single power:

$\frac{2^5 \cdot (4^2)^3 \cdot 8}{2^8 \cdot 16}$

Remember that to work with the laws of indices for the same base, you must first express the numbers $4$, $8$ and $16$ as powers of base $2$.

In computer science, units of measurement increase in powers of base $2$. We know that $1 \text{ Kilobyte (KB)} = 2^{10} \text{ bytes}$. If a server at a school has a storage capacity of $64 \text{ KB}$ and we want to expand it by installing $4$ additional memory cards, each with a capacity of $16 \text{ KB}$:

1. a) Express the final total capacity of the server as a single power of base $2$.
2. b) How many times larger is the final capacity compared to the initial capacity? (Express your answer as a power).

Calculate the value of the following combined operations expression, respecting the order of operations and applying the laws of indices wherever possible:

$10^2 : [5^2 \cdot (2^5 : 2^3)] + \sqrt{144} \cdot (3^2 - 2^3)$

An astronomer observes two galaxies. Galaxy A is at a distance of $3^8 \text{ km}$ from Earth and Galaxy B is at a distance of $27^3 \text{ km}$.

1. a) Which of the two galaxies is further away from us? Justify your answer using powers with the same base.
2. b) If light travels approximately $3 \cdot 10^5 \text{ km}$ in one second, how many seconds would it take for light from Galaxy A to reach Earth? (Leave your answer expressed as a power operation).

Find the value of the unknown index $x$ in the following equations by applying the laws of indices:

1. a) $3^x \cdot 9^2 = 3^7$
2. b) $(5^3)^x : 125 = 5^9$
3. c) $2^4 \cdot 4^x \cdot 8 = 2^{11}$

A TikTok video goes viral. Initially, $2$ people share it. With each passing hour, the number of people who have shared it doubles.

1. a) How many people will have shared the video after $10$ hours? Express your answer as a power and as a natural number.
2. b) If we want to represent the total number of people in standard form after $20$ hours, what would the approximate value be if $2^{10} \approx 10^3$?

A square courtyard is paved with tiles that are also square. The total number of tiles is the result of the expression:

$N = \sqrt{5^2 - 3^2} \cdot \sqrt{25 \cdot 4 \cdot 10^2}$

1. a) Calculate the value of $N$ (total number of tiles).
2. b) How many tiles are on each side of the courtyard?
3. c) If each tile measures $40 \text{ cm}$ on a side, what is the total area of the courtyard in square metres?

Use prime factorisation to calculate the exact square root of the following numbers in a simplified way:

1. a) $\sqrt{2^4 \cdot 3^6 \cdot 5^2}$
2. b) $\sqrt{1296}$ (Hint: factorise the number first).
3. c) $\sqrt{\frac{10^4 \cdot 2^2}{25}}$

In a marine research centre, the cell reproduction of a microscopic alga is being analysed. The team models the process and obtains the following equality with two unknown exponents, $a$ and $b$:

1. a) Decompose the numbers $4$ and $8$ into powers of base $2$ and apply the power of a power, product of powers with the same base, and quotient of powers with the same base properties to reduce the entire left-hand side to a single power of base $2$.
2. b) Deduce the numerical relationship that the exponents $a$ and $b$ must satisfy for the equality to hold.
3. c) Find two different pairs of natural numbers $(a, b)$, both greater than zero, that satisfy this condition. Choose one of the two pairs and verify that the equality holds step by step.

We want to find out which of these two numbers is greater without calculating their gigantic numerical expansions: $A = 2^{60}$ and $B = 3^{40}$.

In class, two different viewpoints arise:

• David claims: "$A$ is greater than $B$ because its exponent is $60$ and $B$'s is only $40$. The greater the exponent, the greater the result always is".

• Sara refutes him: "That is not always true when the bases are different. However, we can transform both powers using the power of a power property so that they have the same exponent, and then compare them with certainty".

1. a) Prove that David's claim is false by proposing a simple counterexample with small numbers (write two powers where the one with the smaller exponent has a greater numerical value than the one with the greater exponent).
2. b) Help Sara: calculate the greatest common divisor of the exponents $60$ and $40$, and express $2^{60}$ and $3^{40}$ in the form $(a^m)^k$, so that both are raised to the same common exponent $k$.
3. c) Evaluate the inner powers of the bases obtained in the previous section and justify rigorously which of the two original numbers, $A$ or $B$, is the greater.