Indices and standard form – Year 10 GCSE maths worksheet with answers to download as PDF

Non-calculator. Simplify each expression, leaving numbers as powers where sensible.

1. \(3^5\times3^4\) and \(7^8\div7^3\)
2. \((2^3)^4\) and \(5^0\)
3. \(2x^3\times4x^5\)
4. \(\dfrac{12a^7}{3a^2}\) and \((3b^2)^3\)

Non-calculator. Work out the value of each as an integer or a fraction.

1. \(5^0\) and \(10^{-3}\)
2. \(4^{-2}\)
3. \(\left(\dfrac13\right)^{-2}\)
4. \(2^{-1}+2^{-2}\)

Non-calculator. Write down the value of each.

1. \(\sqrt{144}\) and \(\sqrt[3]{125}\)
2. \(2^6\) and \(15^2\)
3. \(\sqrt[3]{-27}\) and \((-4)^3\)
4. \(\sqrt{0.09}\) and \(\sqrt{\dfrac{49}{100}}\)

Write each number in standard form \(a\times10^n\) where \(1\le a\lt 10\). Then write the last two as ordinary numbers.

1. \(4\,500\,000\) and \(83\,000\,000\,000\)
2. \(0.00072\) and \(0.0000605\)
3. \(3.2\times10^4\)
4. \(7.05\times10^{-3}\)

Non-calculator. Give each answer in standard form.

1. \((3\times10^4)\times(2\times10^3)\)
2. \((8\times10^6)\div(2\times10^2)\)
3. \((4.2\times10^5)+(3\times10^4)\)
4. \((6.3\times10^3)\times(5\times10^{-5})\)

Put these numbers in order, smallest first. Compare the powers of 10 first.

1. \(3.2\times10^5\), \(4.1\times10^4\), \(9.8\times10^5\), \(2.5\times10^{-1}\)
2. \(6\times10^{-3}\), \(5.9\times10^{-3}\), \(0.0061\), \(6.01\times10^{-4}\)
3. Which is bigger, \(7.3\times10^8\) or \(2.1\times10^9\)? Explain.
4. Write \(2^{10}\), \(10^3\) and \(3^7\) in order of size, smallest first.

The Sun is \(1.5\times10^8\) km from the Earth. Light travels at \(3\times10^5\) km per second. Calculator allowed.

1. How long does light take to reach the Earth from the Sun, in seconds?
2. Write your answer to part (a) in minutes and seconds.
3. The Moon is \(3.84\times10^5\) km from the Earth. How long does light take from the Moon, to 3 significant figures?
4. How many times further is the Sun than the Moon? Give your answer to the nearest whole number.

The population of the world is about \(8.1\times10^9\) and the population of the UK is about \(6.7\times10^7\). Calculator allowed.

1. How many times bigger is the world population than the UK population? Give your answer to the nearest whole number.
2. What percentage of the world population lives in the UK, to 2 decimal places?
3. The UK’s land area is \(2.4\times10^5\) km². Find the population density in people per km² to the nearest whole number.
4. By 2050 the UK population is expected to grow by 10%. Write the new population in standard form.

A hydrogen atom has a diameter of about \(1.06\times10^{-10}\) metres. Calculator allowed.

1. How many hydrogen atoms in a line would fit across a distance of 1 mm (\(10^{-3}\) m)? Give your answer in standard form to 3 s.f.
2. A grain of sand has a diameter of \(2\times10^{-4}\) m. How many atoms would fit across the grain?
3. A human hair has a width of \(7\times10^{-5}\) m. How many times wider is the hair than one atom, to 2 s.f.?
4. Write the diameter of the atom in nanometres, using \(1\text{ nm}=10^{-9}\) m.

One terabyte (1 TB) is \(10^{12}\) bytes. A photo needs about \(4\times10^6\) bytes and a song needs about \(5\times10^6\) bytes. Calculator allowed.

1. How many photos fit on a 1 TB drive?
2. How many songs fit on a 1 TB drive?
3. A drive holds 2000 songs and 50 000 photos. How many bytes is that, in standard form?
4. What fraction of 1 TB is used in part (c)? Give your answer as a percentage.

A type of bacterium is \(2.5\times10^{-6}\) m long. Calculator allowed.

1. How many of these bacteria placed end to end would stretch 1 metre?
2. They are placed in a line along the length of a 15 cm ruler. How many bacteria is that?
3. The bacteria double in number every 20 minutes. Starting with 500, how many are there after 6 hours? Give your answer in standard form to 3 s.f.
4. How many hours until there are more than \(10^{12}\) bacteria?

A sheet of paper is 0.1 mm thick. Each fold doubles the thickness.

1. Write a formula for the thickness \(T\) mm after \(n\) folds.
2. Find the thickness after 10 folds.
3. Find the thickness after 20 folds in metres, to 3 significant figures, in standard form.
4. How many folds are needed to get thicker than the height of a 2 m door?

At one time the UK national debt was about £\(2.7\times10^{12}\). The UK population was about \(6.7\times10^7\). Calculator allowed.

1. Work out the debt per person, in standard form to 2 s.f.
2. A family of four has a yearly income of £\(6.4\times10^4\). How many years of their income equals their share of the debt?
3. Each year the debt grows by 2.5%. By how many pounds does it grow in the first year? Give your answer in standard form.
4. What is the new debt after the first year, in standard form to 3 s.f.?

A wind farm produces \(1.8\times10^7\) watts. A kettle uses \(2.4\times10^3\) watts. Calculator allowed.

1. How many kettles could the wind farm run at the same time?
2. The farm runs for \(8\times10^3\) hours in a year. How many watt-hours does it produce? Give your answer in standard form.
3. A house uses \(3.2\times10^3\) kilowatt-hours per year. If 1 kilowatt-hour is \(10^3\) watt-hours, how many houses could be supplied by the farm’s yearly output?
4. Write the yearly output in kilowatt-hours.

Higher tier. Use \(a^{\frac1n}=\sqrt[n]a\) and \(a^{\frac mn}=\left(\sqrt[n]a\right)^m\). Non-calculator.

1. \(25^{\frac12}\) and \(27^{\frac13}\)
2. \(8^{\frac23}\) and \(16^{\frac34}\)
3. \(27^{-\frac23}\) and \(\left(\dfrac14\right)^{-\frac32}\)
4. Solve \(2^x=\dfrac1{32}\) and \(9^x=27\).

Use the index laws to prove each statement for any whole number \(n\).

1. Show that \(2^n+2^{n+1}=3\times2^n\).
2. Show that \(5^{n+2}-5^n=24\times5^n\).
3. Show that \(4^{n+1}\) is a multiple of 4.
4. Show that \(\dfrac{6^{n+1}}{2^n}=6\times3^n\).

Write both sides as powers of the same base, then compare the indices.

1. \(3^{x+1}=81\)
2. \(2^{2x}=64\)
3. \(5^x=\dfrac1{25}\)
4. Higher tier: \(4^x=8\).

Higher tier. Round each number to 1 significant figure to estimate. Then use a calculator and find the percentage error of your estimate to 1 d.p.

1. Estimate \(\dfrac{4.8\times19.7}{0.21}\).
2. Estimate \(\sqrt{\dfrac{9.2\times51}{1.9}}\).
3. Work out the accurate value of (a) to 2 d.p. and the percentage error of your estimate.
4. Explain whether your estimate for (a) is an overestimate or underestimate, and why.

The mass of the Sun is \(1.99\times10^{30}\) kg and the mass of the Earth is \(5.97\times10^{24}\) kg. Jupiter’s mass is \(1.9\times10^{27}\) kg.

1. How many times heavier is the Sun than the Earth, in standard form to 3 s.f.?
2. How many times heavier is Jupiter than the Earth, to 3 s.f.?
3. What fraction of the Sun’s mass is Jupiter, in standard form to 2 s.f.?
4. Find the combined mass of the Earth and Jupiter in standard form to 3 s.f.

Calculator allowed.

1. Which is bigger, \(2^{100}\) or \(10^{30}\)? Use \(2^{10}=1024\) to explain.
2. How many digits does \(2^{100}\) have?
3. The last digits of powers of 7 repeat: 7, 9, 3, 1, ... What is the last digit of \(7^{2025}\)?
4. Show that \(2^{10}\approx10^3\) and use it to estimate \(2^{40}\) in standard form.