Number worksheet — GCSE Higher

1.Write 200 as a product of its prime factors, using index notation.

2.Expand and simplify (2 + √3)², giving your answer in the form a + b√3.

3.A circle has a radius of 3 cm. Which of these is the exact area of the circle?

4.Given that 2⁵ = 32, work out 2⁶.

5.Write 7/12 as a decimal, showing clearly which digit is recurring.

6.Write 90 as a product of its prime factors.

7.Four students each estimate √70 without a calculator. Ben says: '8² = 64 and 9² = 81, and 70 is about a third of the way from 64 to 81, so √70 ≈ 8.3.' Chloe says: '8.5² = 72.25, which is close to 70 but a little too big, so I'll bring it down slightly to √70 ≈ 8.4.' Dan says: '9² = 81 is closer to 70 than 8² = 64 is, so √70 ≈ 8.9.' Ella says: '8² = 64 and 9² = 81, so I'll just take the midpoint of 8 and 9: √70 ≈ 8.5.' Given that √70 = 8.3666... to 4 decimal places, whose estimate is closest?

8.In a school long jump competition Priya's jump is recorded as 3.8 m and Nadia's jump is recorded as 3.9 m, each correct to the nearest 0.1 m. Priya says she may have jumped further than Nadia. Decide whether Priya is right, and give a reason for your answer.

9.Work out (−36) ÷ (−6) × 2

10.The decimal 0.2333... has one non-recurring digit (the 2) followed by a single recurring digit (the 3), so it can be written as 0.2 recurring 3. Let x = 0.2333... . Work out x as a fraction in its simplest form.

11.Work out 36 ÷ (2 × 3)

12.The population of a planet is 8 × 10⁹. Write this population as an ordinary number.

13.Decide which of 2³⁰ and 3²⁰ is the larger number, and write down the correct statement.

14.During one night on a mountain, four hikers each recorded the temperature they felt: Ben −6 °C, Priya −2 °C, Sam −9 °C, Alex 1 °C. Work out the difference between the coldest and the warmest of these four temperatures.

15.Write 5/6 as a decimal, showing clearly which digit recurs.