Properties of Powers

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) $2024^{0} =$

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) $\sqrt{100} =$

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 $2$ meters for a gate on one side.

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

A square patio is paved with square tiles. The total number of tiles is the result of the expression:

$N = \sqrt{4^2 + 3^2} \times \sqrt{25 \times 4 \times 10^2}$

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

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

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

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) $3^{2} \times 3^{4} =$
2. b) $5^{3} \times 5^{3} =$
3. c) $a^{2} \times a^{5} =$
4. d) $10^{2} \times 10 \times 10^{3} =$ (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) $m^{10} : m^{9} =$

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.

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) $(x^{3})^{4} =$

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.

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) $(2 \times 5)^3$
2. b) $(12 : 3)^2$
3. c) $(2^2 \times 3)^2$

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

1. a) $5^x \times 5^4 = 5^9$
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).

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

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

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 $3 \times 10^5 \text{ km}$ 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).

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

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

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

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

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

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

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 $2$.

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.

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).

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 $2^{10} \approx 10^3$?