Powers, roots and standard form – Year 8 maths worksheet with answers to download as PDF

Learn your squares, cubes and roots. Work these out without a calculator.

1. \(2^5\)
2. \(3^4\)
3. \(\sqrt{169}\)
4. \(\sqrt[3]{64}\)
5. \(10^6\)

Write down the two whole numbers between which \(\sqrt{75}\) lies, and estimate its value.

Write each answer as a single power. Then, where possible, find its value.

1. \(2^3\times2^4\)
2. \(5^9\div5^6\)
3. \((3^2)^3\)
4. \(\dfrac{7^5\times7^2}{7^4}\)
5. \(4^0\)

Work out the value of each, giving fractions where needed.

1. \(2^{-3}\)
2. \(10^{-2}\)
3. \(5^{-1}\)
4. \(\left(\dfrac12\right)^{-2}\)
5. \(3^{-2}\times3^4\)

Use factor trees to write each number as a product of prime factors (in index form).

1. 60
2. 72
3. 180
4. Use your answers to find the highest common factor (HCF) of 60 and 72.
5. Find the lowest common multiple (LCM) of 60 and 72.

Write each number in standard form.

1. 45 000
2. 7 200 000
3. 0.0006
4. 0.000 034 5
5. 500

Which of these is NOT in standard form? Explain: \(0.8\times10^5\), \(3.2\times10^{-4}\), \(12\times10^3\).

Work these out and give each answer in standard form.

1. \((3\times10^4)\times(2\times10^3)\)
2. \((8\times10^7)\div(4\times10^2)\)
3. \((6\times10^5)\times(5\times10^4)\)
4. \(4.5\times10^3+6\times10^2\)

The population of the United Kingdom is about 67 000 000. The population of London is about 9 000 000.

1. Write each population in standard form.
2. Work out the number of people in the UK who do not live in London, in standard form.
3. What fraction of the UK population lives in London? Give a decimal to 2 decimal places.
4. The UK has an area of about \(2.4\times10^{5}\) km². Work out the average number of people per square kilometre, to the nearest whole number.

Space scientists use standard form all the time. The Sun is about 150 million km from Earth. The Moon is about 384 000 km from Earth. Light travels at \(3\times10^{5}\) km per second.

1. Write both distances in standard form.
2. How many times further away is the Sun than the Moon? Give your answer to the nearest whole number.
3. How long does light take to travel from the Sun to the Earth? Give your answer in seconds and in minutes.
4. How long does light take from the Moon to Earth, to 1 decimal place of a second?

A scientist grows bacteria in a dish. At 9 am there is 1 bacterium. The number doubles every 20 minutes.

1. Complete the table for 10:00 and 10:20 using powers of 2.
2. How many bacteria are there at 12 noon? Write as a power of 2 and as a number.
3. How many hours will it take for the number to pass 1 million? (Use \(2^{20}\approx1\,048\,576\).)
4. Write your answer to part (b) in standard form.

Computer memory is measured in bytes. 1 kilobyte (KB) is \(2^{10}\) bytes, and 1 megabyte (MB) is \(2^{10}\) kilobytes.

1. How many bytes are in 1 KB?
2. Write the number of bytes in 1 MB as a power of 2 and as a number.
3. A phone has 64 GB of storage, where 1 GB is \(2^{10}\) MB. Write 64 GB in bytes as a power of 2.
4. A photo is about \(2^{22}\) bytes. How many photos fit on the phone? Use the index laws.

A sheet of paper is 0.1 mm thick. Priya folds it in half again and again. Each fold doubles the thickness.

1. Complete the row for 3 folds.
2. How many layers after 10 folds? Write as a power of 2.
3. Find the thickness after 10 folds in mm and in cm.
4. How many folds would be needed to make a stack taller than a 1.8 m door? (Use a calculator and work in mm.)

A light year is the distance that light travels in one year, about \(9.5\times10^{12}\) km. The nearest star after the Sun, Proxima Centauri, is 4.25 light years away.

1. Work out the distance to Proxima Centauri in km, in standard form to 3 significant figures.
2. A spacecraft travels at \(5\times10^{4}\) km per hour. How many hours would it take to reach Proxima Centauri? Give your answer in standard form.
3. Convert your answer to years (365 days of 24 hours) and round to the nearest thousand years.
4. Explain why standard form is useful for these numbers.

Biologists measure tiny things in metres. A human hair is about 0.00007 m wide. A bacterium is about \(2\times10^{-6}\) m long. A virus is about \(1\times10^{-7}\) m across.

1. Write the width of a human hair in standard form.
2. Put the three sizes in order, smallest first.
3. How many times wider than a virus is a human hair?
4. How many bacteria laid end to end would match the width of a hair?

Legend says that a clever pupil asked the king for 1 grain of rice on the first square of a chessboard, 2 on the second, 4 on the third and so on, doubling each time.

1. How many grains are on the 6th square? Write as a power of 2.
2. How many grains are on the last (64th) square? Give your answer as a power of 2.
3. Use a calculator to write \(2^{63}\) in standard form to 3 significant figures.
4. About 50 grains of rice weigh 1 g. Find the mass of the grains on the last square in tonnes, in standard form (1 tonne = \(10^6\) g).

The table shows the population and area of four countries.

1. Work out the population density (people per km²) of each country to the nearest whole number.
2. Which country is the most crowded and which is the least crowded?
3. How many times more crowded is Singapore than the UK? Round to the nearest whole number.
4. If the UK population grows by 1.2% to the nearest thousand, write the new population in standard form.

Two school bells ring at the start of lessons. Bell A rings every 12 minutes and bell B rings every 18 minutes. They ring together at 9:00 am.

1. Write 12 and 18 as products of prime factors.
2. Find the LCM of 12 and 18.
3. At what time do the bells next ring together?
4. How many times between 9:00 am and 12:00 noon (including 9:00) do they ring together?
5. Find the HCF of 12 and 18 and explain what it tells you.

Four pupils each simplified the same expression, \(\dfrac{2^6\times2^3}{2^4}\). Their answers are shown.

1. Work out the correct answer as a power of 2 and as a number.
2. Which pupil is correct? Find the mistake in each of the other three answers.
3. Evaluate \(2^6\times2^3\) and \(2^4\) separately and divide to check.
4. Simplify \(\dfrac{5^8\times5^{-3}}{5^2}\).

A factory makes tiny screws. Each screw has a mass of \(2.5\times10^{-3}\) kg. The factory makes \(4\times10^{6}\) screws per day.

1. Find the total mass of one day's screws, in kg and in standard form.
2. Write the mass in tonnes (1 tonne = \(10^{3}\) kg).
3. A lorry can carry \(8\times10^{3}\) kg. How many lorry loads are needed for one day's screws?
4. Over 5 working days, what is the total mass in standard form?

A board game uses square and cube numbers. A player moves along a path of squares numbered 1 to 100. A token lands on every square number, and a bonus token lands on every cube number.

1. List the square numbers up to 100 and the cube numbers up to 100.
2. Which numbers between 1 and 100 are both square and cube? Use powers to explain.
3. How many squares have a token and how many have a bonus token?
4. A player lands on a square that is a cube and a square, and then moves forward \(\sqrt{64}\) squares. Where does she end up if she started at 64?

Estimate how many seconds there are in a lifetime of 80 years, and then compare it with the number of heartbeats.

1. Work out the number of seconds in one year (365 days) and write it in standard form to 3 significant figures.
2. Find the number of seconds in 80 years, in standard form to 2 significant figures.
3. A heart beats about 70 times per minute. How many beats in 80 years? Write in standard form to 2 s.f.
4. If you spent \(\dfrac{1}{3}\) of your life asleep, how many seconds is that? Give your answer in standard form to 2 s.f.