Powers and Exponents

Express the following products as a power and indicate which is the base and which is the exponent in each case:

1. a) $5 \cdot 5 \cdot 5 \cdot 5$

1. b) $12 \cdot 12 \cdot 12$

1. c) $2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2$

1. d) $10 \cdot 10 \cdot 10 \cdot 10 \cdot 10$

Calculate the value of the following basic powers:

1. a) $3^{4}$

1. b) $2^{5}$

1. c) $7^{2}$

1. d) $11^{0}$

1. e) $1^{25}$

Look at the following square made up of small cells. Express the total number of cells as a power and calculate its result.

A content creator publishes a video that goes viral. If every hour the number of people who have seen it triples ($3^{n}$), complete the following table expressing the results as a power:

Time elapsed	People who have seen the video
At the start (Hour 0)	$3^{0} = 1$
After 1 hour	$3^{1} = 3$
After 2 hours
After 5 hours
After $n$ hours

A video by a streamer has gone viral on a social network. During the first few hours, the number of people sharing the video triples every hour. If 3 people shared it at the beginning:

a) Write as a power the number of people who will have shared the video after 5 hours.

b) Calculate the value of this power to find the total number of people at that time.

Apply the properties of powers to write the result as a single power:

1. a) $3^{4} \cdot 3^{2} \cdot 3^{5}$

1. b) $a^{3} \cdot a^{4}$

1. c) $10^{2} \cdot 10 \cdot 10^{3}$

1. d) $2^{6} \cdot 2^{6}$

Reduce to a single power using the property of the quotient of powers with the same base:

1. a) $5^{8} : 5^{3}$

1. b) $x^{10} : x^{7}$

1. c) $10^{6} : 10^{5}$

1. d) $\dfrac{2^{9}}{2^{4}}$

Apply the properties of powers to reduce the following expressions to a single power:

a) $2^5 \cdot 2^3 \cdot 2 =$

b) $(3^4)^3 : 3^7 =$

c) $\frac{5^8 \cdot 5^2}{5^7} =$

Simplify the following expressions, leaving the result as a single power:

1. a) $(4^{3})^{2}$

1. b) $(10^{5})^{4}$

1. c) $(x^{2})^{8}$

In a logistics warehouse, boxes are organized on cubic shelves. A storage block is made up of 4 shelves in width, 4 in length, and 4 in height. If each shelf contains 4 boxes:

a) Express the total number of boxes as a single power.

b) If each box contains $2^3$ packs of paper, how many packs of paper are there in total in the block? Express the result as a power of base 2.

In an archaeological excavation near ancient Mesopotamia, a ceremonial structure made of cubic stone blocks has been found. The structure follows a pattern of "nested cubes". The first level is a cube of $2 \cdot 2 \cdot 2$ blocks. The second level surrounds the previous one forming a cube of $4 \cdot 4 \cdot 4$ blocks, and the third one of $8 \cdot 8 \cdot 8$ blocks.

1. a) Express as a single power of base 2 the total number of blocks there would be on the fifth level if the edge of the cube continues to double at each step.
2. b) If each block has a mass of $2^6$ pounds, express the total mass of the fifth-level structure as a single power.

Write the polynomial expansion of the following numbers using powers of base $10$:

1. a) $4\,527$

1. b) $30\,816$

1. c) $504$

Write the polynomial expansion of the following numbers using powers of base 10:

a) $45\,208$

b) $306\,750$

c) $7\,003\,040$

Express the following data using standard form:

a) The average distance from the Earth to the Moon is about $384\,400$ miles.

b) The approximate world population is $8\,000\,000\,000$ people.

c) A light year is approximately equal to $9\,461\,000\,000\,000$ miles.

The storage of a supercomputer is measured in "Exabytes". Knowing that $1 \text{ kilobyte} = 10^3 \text{ bytes}$, $1 \text{ megabyte} = 10^3 \text{ kilobytes}$, $1 \text{ gigabyte} = 10^3 \text{ megabytes}$ and so on following the powers of 10 (terabyte, petabyte, exabyte):

1. a) Express in scientific notation how many bytes there are in an Exabyte.
2. b) If a high-definition video file takes up $2 \cdot 10^9$ bytes, how many files of this type can fit on a $4 \text{ Petabytes}$ disk? Express the result in scientific notation.

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

a) $7^x \cdot 7^2 = 7^9$

b) $(5^3)^x = 5^{15}$

c) $10^8 : 10^x = 10^3$

Find the value of the unknown $n$ in the following equalities, applying the rules of powers:

1. a) $2^n \cdot 2^3 \cdot 2 = 1024$
2. b) $5^6 : 5^n = 125$
3. c) $(3^2)^n \cdot 9 = 3^{10}$

Which of the following expressions is greater? Perform the necessary calculations to justify your answer without using a calculator.

a) $2^6$

b) $4^3$

c) $8^2$

d) $64^1$

What relationship do you find between the bases of all these powers?

Without calculating the full value of the powers, give a reasoned answer as to which of the following numbers is greater. To do this, use the transformation of the bases to a common base:

$$ A = 8^{20} \cdot 4^{15} $$ $$ B = 16^{18} \cdot 2^5 $$

A new social network has detected that the number of "shares" of a fake news story triples every hour. At 10:00 a.m., the news has already been shared $3^4$ times.

1. a) At what time will the number of shares exceed $3^{10}$ for the first time?
2. b) If the server crashes when the number of shares reaches $9^7$, how many hours will have passed from 10:00 a.m. until the crash? Express all the steps using the properties of powers.

Calculate the value of the following combined expressions respecting the order of operations:

a) $2^4 - 3^2 + \sqrt{25} \cdot 2 =$

b) $(5-2)^3 + 10^2 : (2^2 + 1) =$

c) $2 \cdot (3^2 - 4) + \sqrt{64} : 2^3 =$

A square mosaic made up of square tiles is to be designed. It is known that the number of tiles used is the solution to the following operation:

$$ \sqrt{10^2 - 8^2} \cdot (3^3 - 5^2)^2 - 2^3 $$

1. a) How many tiles will the mosaic have in total?
2. b) How many tiles will there be on each side of the square mosaic?

Calculate the value of the following combined expression, applying the properties of powers whenever possible to simplify the calculation before working it out:

$$ \frac{(2^3 \cdot 2^2)^2 \cdot 5^7 \cdot 10^3}{(5^2)^4 \cdot 2^{12}} + \sqrt{144} : 2^2 $$

A natural number $X$ has the following prime factorization: $X = 2^a \cdot 3^2 \cdot 5^b$.

It is known that:

• The greatest common factor of $X$ and $180$ is $2^2 \cdot 3^2 \cdot 5^0$.

• The least common multiple of $X$ and $180$ is $2^3 \cdot 3^2 \cdot 5^1$.

Find the values of the exponents $a$ and $b$ and write the number $X$ in decimal notation. (Fact: $180 = 2^2 \cdot 3^2 \cdot 5^1$).