IGCSE · Practice questions · Squares, cubes and roots

Squares to 15, by heart.

Six original Cambridge-style questions on squares, cubes, square roots, cube roots and other powers, with and without a calculator.

Original questions All questions on this page are original work, written in the Cambridge IGCSE style. They are not from past papers. They test the same concepts and skills the syllabus rewards.

The method

Know the facts, then let the root undo the power.

Recall: squares 1 to 15 (up to 225); cubes of 1, 2, 3, 4, 5 and 10.
Decimals: \(0.3^{2} = 0.09\) (2 d.p.), \(0.2^{3} = 0.008\) (3 d.p.).
Signs: \(\sqrt[3]{-64} = -4\); \((-3)^{2} = 9\) but \(-3^{2} = -9\).
Under a root: work out everything underneath first; the root does not split over \(+\) or \(-\).
Trapping: find the squares (or cubes) either side to locate a root.

Recall

[3 marks]

Write down the value of (a) \(\sqrt{196}\) (b) \(\sqrt[3]{125}\) (c) \(11^{2}\).

(a) 14 ✓
(b) 5 ✓
(c) 121 ✓

Recall

[2 marks]

Without a calculator, work out (a) \(2^{5}\) (b) \(\sqrt[3]{-1000}\).

(a) \(2 \times 2 \times 2 \times 2 \times 2 = 32\) ✓
(b) \((-10)^{3} = -1000\), so \(-10\) ✓

Calculation

[2 marks]

Without a calculator, work out \(\sqrt{0.09} + 0.4^{2}\).

\(\sqrt{0.09} = 0.3\) and \(0.4^{2} = 0.16\) ✓
\(0.3 + 0.16 = 0.46\) ✓

Calculation

[2 marks]

Use your calculator to work out \(\sqrt[3]{\dfrac{9.8^{2} - 6.1}{2.3}}\). Give your answer correct to 3 significant figures.

\(9.8^{2} - 6.1 = 96.04 - 6.1 = 89.94\) ✓
\(89.94 \div 2.3 = 39.104\dots\) ✓
\(\sqrt[3]{39.104\dots} = 3.394\dots = 3.39\) ✓

Analysis

[3 marks]

Without a calculator: (a) Between which two consecutive integers does \(\sqrt{90}\) lie? (b) Show that \(\sqrt{60}\) is larger than \(\sqrt[3]{400}\).

(a) \(9^{2} = 81\) and \(10^{2} = 100\), so \(9 < \sqrt{90} < 10\). ✓
(b) \(7.5^{2} = 56.25 < 60\), so \(\sqrt{60} > 7.5\). ✓
\(7.5^{3} = 421.875 > 400\), so \(\sqrt[3]{400} < 7.5\). Hence \(\sqrt{60} > \sqrt[3]{400}\). ✓

Analysis

[3 marks]

A water tank is a cube with volume 1.728 m³. It stands in the middle of a square concrete slab of area 2.56 m², leaving a border of equal width all round.

Find the width of the border.

Edge of tank \(= \sqrt[3]{1.728} = 1.2\) m (since \(12^{3} = 1728\)). ✓
Side of slab \(= \sqrt{2.56} = 1.6\) m (since \(16^{2} = 256\)). ✓
Border \(= (1.6 - 1.2) \div 2 = 0.2\) m. ✓

Mark this once you have attempted all six and checked your working. It records a Practiced badge on the topic and adds a one-time bonus. Revealing the solutions alone does not count.