Exercises: Powers

Express the following products as a power and state the base and 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) $10 \cdot 10 \cdot 10 \cdot 10 \cdot 10$

Work out 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 of small cells. Express the total number of cells as a power and work out the result.

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

1. a) $4\,527$

1. b) $30\,816$

1. c) $504$

Apply the laws of indices 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) $2^{6} \cdot 2^{6}$

Simplify to a single power using the division law of indices for 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) $\dfrac{2^{9}}{2^{4}}$

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

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

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

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

A content creator posts a video that goes viral. If the number of views triples every hour ($3^{n}$), complete the following table, expressing the results as powers:

Time elapsed	People who have watched 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

In a 2nd year ESO exam, Lucas solved two simplification exercises, making common conceptual errors:

• Case A: He wrote that $3^2 \cdot 3^4 = 9^6$, justifying that "bases are multiplied and exponents are added".

• Case B: He wrote that $2^3 \cdot 5^3 = 10^6$, justifying that "bases are multiplied and exponents are added".

1. a) Explain with reasons the error made in Case A, indicate which property he should have applied, and write the correct result as a single power.
2. b) Explain the error made in Case B, which property corresponds to that operation, and calculate the correct power.
3. c) Write a clear and brief rule so that a classmate does not confuse when exponents should be added and when bases should be multiplied.

Solve the following challenges by constructing powers that satisfy the required conditions:

1. a) Find three different ways to write the number $64$ as a single power $a^b$, where both the base $a$ and the exponent $b$ are natural numbers greater than $1$.
2. b) Write the number $81$ as a power with base $3$ and as a power with base $9$. Show, by applying the power of a power property, why both expressions represent exactly the same quantity.
3. c) Write a multiplication of two powers that have different bases and different exponents (both greater than $1$) whose final result is exactly equal to $2^7$.

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 start:

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

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

Apply the laws of indices to simplify the following expressions to a single power:

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

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

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

In a logistics warehouse, boxes are organised into cubic shelving units. A storage block consists of 4 shelving units in width, 4 in length, and 4 in height. If each shelving unit 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.

Express the following data using standard form:

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

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

c) One light-year is approximately equal to $9\,461\,000\,000\,000$ kilometres.

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

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

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

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

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

a) $45\,208$

b) $306\,750$

c) $7\,003\,040$

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

Which of the following expressions is larger? Show 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?

To simplify and calculate the numerical expression $\dfrac{12^4}{3^4 \cdot 4^2}$ without using a calculator, two students propose different strategies:

• Lucía's Method: She first groups the numerator and part of the denominator using the power of a quotient: $\dfrac{12^4}{3^4} = (12 : 3)^4 = 4^4$. Then, she divides $4^4 : 4^2$ by subtracting exponents.

• Marcos's Method: He breaks down the number $12$ into prime factors ($12 = 2^2 \cdot 3$), applies the power of a product in the numerator, breaks down the denominator ($4 = 2^2$), and simplifies by subtracting exponents of powers with the same base.

1. a) Work through the procedure step by step following Lucía's Method until finding the final value.
2. b) Work through the procedure following Marcos's Method until obtaining the result.
3. c) Compare both paths: which one do you consider faster in this exercise and why? In what situation could Lucía's method not be applied and Marcos's would be essential?

Faced with the heavy snowfall warning issued by AEMET in a mountain region, Civil Protection launches a chain protocol to alert isolated shelters:

• In step 1 (10 minutes after the alert), the regional centre alerts 3 main posts.

• In step 2 (at 20 minutes), each of those 3 posts alerts 3 zone shelters.

• In each new 10-minute step, each shelter alerted in the previous round contacts 3 new local shelters, without interrupting the chain.

1. a) Model using a power of base 3 the number of new shelters that receive the alert exactly in step $n$. How many are alerted in step 4?
2. b) Express as a sum of powers the cumulative total of shelters alerted from the beginning up to the end of step 3 inclusive.
3. c) If the chain ends when in a certain step $3^6$ shelters are alerted, how many minutes have passed in total since the first alert started? Justify how you obtain the time from the exponent.

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 level forms one of $8 \cdot 8 \cdot 8$ blocks.

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

A new social network has detected that the number of "shares" of a fake news story triples every hour. At 10:00 am, the story 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 am until the crash? Express all steps using the laws of indices.

Work out the value of the following combined expression, applying the laws of indices whenever possible to simplify the calculation before computing:

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

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 powers of 10 (terabyte, petabyte, exabyte):

1. a) Express in standard form 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 fit onto a disk of $4 \text{ Petabytes}$? Express the result in standard form.

Without calculating the powers in full, state with reasons which of the following numbers is larger. To do this, rewrite the bases using a common base:

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

Find the value of the unknown $n$ in the following equations, applying the laws of indices:

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

A square mosaic made 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 along each side of the square mosaic?

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 divisor 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. (Hint: $180 = 2^2 \cdot 3^2 \cdot 5^1$).

The archive of a Spanish town hall is going to digitise historical records. They have catalogued $2^9$ boxes; each box contains $2^6$ bundles of documents and each bundle occupies, scanned in high resolution, $2^7\text{ KB}$ (kilobytes).

The technical service is hesitating between purchasing a $64\text{ GB}$ USB flash drive or contracting a $1\text{ TB}$ network drive.

Approximate equivalence data: $1\text{ GB} \approx 10^6\text{ KB}$, $1\text{ TB} \approx 10^9\text{ KB}$, and $2^{10} = 1024 \approx 10^3\text{ KB}$.

1. a) Estimate: Without calculating the detailed numerical value, mentally calculate the total size by adding the exponents in base 2 and use the approximation $2^{10} \approx 10^3$ to estimate the order of magnitude of the archive in powers of 10. State a hypothesis: will the $64\text{ GB}$ flash drive be enough, or will the $1\text{ TB}$ drive be necessary?
2. b) Check: Express the exact total size of the archive as a single power of base 2 applying the properties of the product of powers.
3. c) Break down this power in the form $(2^{10})^2 \cdot 2^k$ to calculate its numerical value in kilobytes and reasonably determine whether your initial estimate was correct and which storage medium is best to buy to avoid overspending.

To decipher a security code, you must find a natural number $x$ that satisfies the following equality with powers:

$$\frac{4^{x+2} \cdot 8^x}{16^x} = 64$$

1. a) Break down the bases $4$, $8$, $16$ and the result $64$ into powers of base $2$.
2. b) Transform the left-hand side into a single power of base 2 by applying in order the properties of the power of a power, and the product and quotient of powers with the same base.
3. c) Equate the exponents obtained on both sides to set up a linear relationship and calculate the value of $x$. Check that your solution is correct by substituting the found value into the initial expression.
4. d) Reverse reasoning: If the right-hand side were $1$ instead of $64$, would there be any natural number $x$ that satisfied the equality? Justify your answer using the property of the zero exponent ($a^0 = 1$ for $a \neq 0$).