Exact calculation: fractions, surds and π worksheet — GCSE Higher

1.A square tile has an area of 72 cm². Work out the exact perimeter of the tile, giving your answer in the form k√2 cm.

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

3.Simplify √45.

4.A student works out the exact area of a circle with radius 4 cm by squaring the radius but forgetting to multiply by π. Work out the correct exact area of the circle, in terms of π.

5.Work out √3 × √12, giving your answer as an integer.

6.A circle has a circumference of 8π cm. Work out the exact area of the circle in terms of π.

7.Rationalise the denominator of 6/√3, giving your answer in its simplest form.

8.Work out 3/4 − 5/12 exactly, giving your answer in its simplest form.

9.Simplify √8 + √18, giving your answer in the form k√2.

10.In a school, 3/5 of the students study French. Of these students, 2/3 also study Spanish. What fraction of all the students study both French and Spanish?

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

12.Rationalise the denominator of 10/(4 − √6), giving your answer in its simplest form.

13.The two shorter sides of a right-angled triangle are √12 cm and √24 cm. Work out the exact length of the hypotenuse.

14.Chloe wants to work out 3 1/4 − 1 2/3. She converts both mixed numbers to twelfths, then subtracts the whole numbers and the fraction parts separately, without checking whether she needs to exchange first. Work out the correct value of 3 1/4 − 1 2/3, giving your answer as a mixed number in its simplest form.

15.Work out 1 1/2 ÷ 3/4 exactly, giving your answer in its simplest form.