GCSE · Integers, powers & roots

Powers and roots (squares, cubes, square roots and cube roots)Exam-style questions

What squares, cubes and other powers mean and how to work them out by hand; square roots and cube roots as the way back, including the negative square root and the cube root of a negative number; powers of 10; trapping a root between two whole numbers and closing in to one decimal place by trial; powers and roots on a calculator; and how to compare and order them.

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11 questions · GCSE (Higher) level · mark scheme and worked solution on each · no sign-up to try

DifficultyAllLevel 2Level 3Level 4
  1. 1
    Show that 70\sqrt{70} is closer to 88 than to 99

    (2 marks)
    Level 4 · 2 marksTry it
  2. 2
    Calculate.

    58.3+4.7231.62\frac{\sqrt[3]{58.3 + 4.7^2}}{1.6^2}

    Give your answer correct to 33 significant figures.

    (2 marks)
    Level 4 · 2 marksTry it
  3. 3
    A water tank is in the shape of a cube.

    When it is full, the tank holds 500500 litres of water.

    11 litre =1000= 1000 cm³

    The tank must fit in the gap between the floor and a shelf.75 cmshelffloorThe gap is 7575 cm high.

    Assume that the walls of the tank have no thickness.

    Will the tank fit in the gap?

    You must show how you get your answer.

    (4 marks)
    Level 4 · 4 marksTry it
  4. 4
    Write these numbers in order of size, starting with the smallest.

    503220036.523\sqrt{50} \qquad 3^2 \qquad \sqrt[3]{200} \qquad 6.5 \qquad 2^3

    Show how you decide.

    (3 marks)
    Level 4 · 3 marksTry it
  5. 5
    (a) pp is a number between 00 and 11

    Circle the expression with the greatest value.

    p3p2ppp^3 \qquad\qquad p^2 \qquad\qquad p \qquad\qquad \sqrt{p}

    (1 mark)
    (b) qq is a number greater than 11

    Circle the expression with the greatest value.

    q3q2qqq^3 \qquad\qquad q^2 \qquad\qquad q \qquad\qquad \sqrt{q}

    (1 mark)
    Level 4 · 2 marksTry it