Properties of Exponents

In an old video rental store, movies were organized on shelves. If we have an expression in the form of a power such as $7^{3}$, indicate which number acts as the base and which as the exponent. Briefly explain what operation you would have to do to calculate its value (without needing to give the final result).

A well-known social network shows users' profile photos in a square grid. If Emma's photo wall 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.

Work out 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) $2024^{0} =$

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

1. a) $\sqrt{16} =$
2. b) $\sqrt{49} =$
3. c) $\sqrt{81} =$
4. d) $\sqrt{100} =$

A school courtyard is square-shaped and has a total area of $225 m^2$. The maintenance team wants to surround it with hazard tape for a sports activity, but leaving a free space of $2$ feet for a door on one of the sides.

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

Apply the property of 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 had a $1$).

Use the property of 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) $m^{10} : m^{9} =$

A courier company organizes its parcels in large cubic containers. If a container holds $6$ boxes in length, $6$ boxes in width and $6$ boxes in 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.

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

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

Use the properties of powers to simplify the following expression and leave it as a single power of base $2$:

Remember to indicate the steps followed: first operate inside the parentheses, then apply the power of a power and, finally, divide.

Calculate the following expressions in two different ways to check that the properties of powers are met:

• Method A: Calculating the inside of the parentheses first and then the power.

• Method B: Applying the property of the power of a product or a quotient.

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

In a video game, every time you pass 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.

In the world of computing, storage capacities are usually expressed in powers of 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 TikTok video goes viral. Initially, $2$ people share it. Every hour that passes, the number of people who have shared it doubles.

1. a) How many people will have shared the video after $10$ hours? Express the result 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 $2^{10} \approx 10^3$?

Apply the properties of powers and make the necessary base transformations to express the result as a single power:

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

Remember that to operate with the properties of the same base, you must first decompose the numbers $4$, $8$ and $16$ into powers of base $2$.

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 of 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 the light from galaxy A to reach Earth? (Leave the result expressed as a power operation).

In the world of computing, units of measurement grow in powers of base $2$. We know that $1 \text{ Kilobyte (KB)} = 2^{10} \text{ bytes}$. If a school server 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 greater is the final capacity compared to the initial one? (Express the result as a power).

Find the value of the letter $x$ in each of the following equalities by applying the properties of powers:

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) $x^0 = 1$ (Indicate which values $x$ could not take in this case).

A TikTok video goes viral. At first, $3$ people share it. Each of those people shares it with $3$ new people, and so on in different levels of diffusion.

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

Find the value of the unknown exponent $x$ in the following equalities by applying the properties of powers:

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

Solve the following combined operations, correctly applying the order of operations and the properties of powers 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) $2^5 : 2^3 + \sqrt{81} \cdot 3 - 1^{15}$

Calculate the value of the following combined expression, respecting the order of operations and applying the properties of powers whenever possible:

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

Use prime factorization 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: factor the number first).
3. c) $\sqrt{\frac{10^4 \cdot 2^2}{25}}$

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

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

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