Powers

Express the following products as a power and state the base and the exponent in each case:

1. a) $5 \times 5 \times 5 \times 5$

1. b) $12 \times 12 \times 12$

1. c) $2 \times 2 \times 2 \times 2 \times 2 \times 2$

1. d) $10 \times 10 \times 10 \times 10 \times 10$

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) $504$

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

a) $45\,208$

b) $306\,750$

c) $7\,003\,040$

Apply the laws of indices to write the result as a single power:

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

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

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

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

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) $\dfrac{2^{9}}{2^{4}}$

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

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

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

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

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 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.

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

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

Work out the value of the following combined expressions respecting the order of operations:

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

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

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

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:

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

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?

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

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

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

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

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?

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

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

Express the following data using scientific notation:

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

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

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

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 $2 \times 10^9$ bytes, how many files of this type fit onto a disk of $4 \text{ Petabytes}$? Express the result in standard form.

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

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.

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.

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 $2 \times 2 \times 2$ blocks. The second level surrounds the previous one, forming a cube of $4 \times 4 \times 4$ blocks, and the third level forms one of $8 \times 8 \times 8$ 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.

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:

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

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

It is known that:

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

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

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