Less common M2.6

Squares Cubes and Roots

This topic covers the essential operations of raising numbers to the power of two (squaring) and three (cubing), along with their inverse operations, finding square and cube roots. A solid grasp of these concepts is fundamental for manipulating numbers and algebraic expressions in more complex problems.

Part of the ESAT Mathematics 1 syllabus — revision for the Engineering and Science Admissions Test (ESAT), the UAT-UK admissions test for Cambridge, Imperial, Oxford and UCL.

Key points

  • Squaring any real number, positive or negative, always yields a positive result. For example, (-5)2 = 25.
  • The square root symbol (sqrt) refers strictly to the positive root. For example, √(36) is 6, not ±6.
  • Cubing a number preserves its original sign. The cube of a positive number is positive, and the cube of a negative number is negative, e.g., (-2)3 = -8.
  • The cube root of a negative number is a real, negative number. For example, the cube root of -125 is -5.

Formulae

x2 = x × x

To find the square of a number x.

y = √(x2)

To find the positive square root of a number. Here, y will equal the absolute value of x.

x3 = x × x × x

To find the cube of a number x.

y = cbrt(x3)

To find the cube root of a number. Here, y will equal x, preserving the sign.

Definitions

Square
The number obtained by multiplying a value by itself. Denoted by a superscript 2.
Positive Square Root
The positive number which, when multiplied by itself, gives the original number. This is the default meaning of the √() symbol.
Negative Square Root
The negative number which, when multiplied by itself, gives the original number.
Cube
The number obtained by multiplying a value by itself twice. Denoted by a superscript 3.
Cube Root
The number which, when cubed, gives the original number. This can be positive or negative.

Worked example

Evaluate: cbrt(-27) + √(16) × (-2)2

  1. 1

    First, evaluate each part of the expression according to the order of operations (powers/roots, then multiplication, then addition).

  2. 2

    Calculate the cube root:

    cbrt(-27) = -3, because (-3) × (-3) × (-3) = -27
  3. 3

    Calculate the square root:

    √(16) = 4

    Remember the √() symbol implies the positive root.

  4. 4

    Calculate the square:

    (-2)2 = (-2) × (-2) = 4

    Note the brackets are important.

  5. 5

    Now substitute these values back into the expression:

    -3 + 4 × 4.

  6. 6

    Perform the multiplication:

    4 × 4 = 16
  7. 7

    Perform the final addition:

    -3 + 16 = 13

Answer: 13

Common mistakes

  • ×Mistaking √(x) for both roots. The expression √(25) equals 5 only. The equation x2 = 25 has two solutions, x = 5 and x = -5, which is a different context.
  • ×Confusing (-x)2 with -x2. For x=4, (-4)2 is 16, whereas -42 is -(4*4) which is -16. Pay close attention to brackets.
  • ×Making sign errors with cubes. Forgetting that the cube of a negative number is negative, e.g., calculating (-3)3 = 27 instead of the correct -27.

No-calculator tips

  • ✓Memorise common powers. Knowing squares up to 152 (225) and cubes up to 53 (125) will save a lot of time.
  • ✓Use the last digit to check roots. A perfect square cannot end in 2, 3, 7, or 8. This can help you quickly eliminate incorrect options.
  • ✓To estimate a square root of a non-square number, find the two closest perfect squares it lies between. For example, √(85) must be between √(81)=9 and √(100)=10.

Read this topic in the official UAT-UK ESAT guide →

All Mathematics 1 topics