Squares Square Roots Cubes and Cube Roots for the TMUA

Updated August 2025

Mastering the properties of squares, cubes, and their roots is essential for the TMUA. This page explains the definitions of these terms, the critical distinction between the positive square root and the negative square root, and how the signs of numbers change or are preserved through these operations.

Core concept

A square ( $x^2$ ) is a number multiplied by itself, while a cube ( $x^3$ ) is a number multiplied by itself twice. The square root ( $\sqrt{x}$ ) is the positive value that yields $x$ when squared, whereas the cube root is the unique value that yields $x$ when cubed.

Understanding Squares

The process of squaring a number involves multiplying that number by itself. In general algebraic terms, the square of a value $x$ is represented as $x^2$ . For example, to find the square of $2.5$ , you calculate $2.5 \times 2.5$ , which equals $6.25$ .

A fundamental property of squares is that the result is always positive, regardless of whether the original number was positive or negative. For instance, squaring the negative number $-4$ results in $(-4) \times (-4) = 16$ . This occurs because the product of two negative values is always positive.

Square Roots and the Negative Square Root

By formal definition, the square root of a number is the positive value that, when multiplied by itself, results in the original number. The symbol $\sqrt{n}$ is used to denote this positive value. For example, the square root of $9$ is written as $\sqrt{9}$ , and therefore $\sqrt{9} = 3$ .

It is vital to distinguish between 'the square root' and 'the negative square root'. While the standard square root is positive, the negative square root refers to the negative value which, when multiplied by itself, produces the original number. For example, while the square root of $9$ is strictly $3$ , we can state that the negative square root of $9$ is $-3$ .

Cube Numbers

The cube of a number is found by multiplying the number by itself and then multiplying that result by the number once more. In general terms, the cube of $x$ is written as $x^3$ . For example, the cube of $1.2$ is calculated as $1.2^3 = 1.2 \times 1.2 \times 1.2 = 1.728$ .

Unlike squares, cubes preserve the sign of the original number. The cube of a positive number remains positive, whereas the cube of a negative number remains negative. This is because a negative multiplied by a negative is positive, and that positive multiplied by a third negative results in a negative value.

Cube Roots

A cube root is a value that, when multiplied by itself and then multiplied by itself again, results in the original number. The sign of a cube root always matches the sign of the number being rooted.

For example, since $4 \times 4 \times 4 = 64$ , the cube root of $64$ is $4$ . If the original number were negative, the cube root would also be negative. This unique relationship means every real number has exactly one real cube root.

Key takeaways

The square of any real number (except zero) is always positive.
The notation $\sqrt{x}$ refers exclusively to the positive square root.
The negative square root is the negative value that, when squared, equals the original number.
Cubes and cube roots preserve the sign of the original value, meaning negative numbers have negative cubes and negative cube roots.

Tips

In TMUA Paper 1, pay close attention to the wording of a question. If a question asks for 'the square root' of $x$ , it is looking for the positive value. If it asks for the values of $a$ such that $a^2 = x$ , you must provide both the positive and negative roots.

Cautions

A common error is forgetting that squaring a negative number results in a positive number. Always use brackets when writing $(-x)^2$ to ensure the negative sign is included in the squaring process.

Insight

This distinction between square and cube roots is the basis for understanding odd and even functions. Squaring is an 'even' operation that loses information about the original sign, while cubing is an 'odd' operation that maintains the original sign, making it a one-to-one mapping.

Worked Examples

Example 1

A positive integer is called a squaresum if and only if it can be written as the sum of the squares of two integers. For example, 61 and 9 are both squaresums since

61 = 5^2 + 6^2

and

9 = 3^2 + 0^2

.
A prime number is called awkward if and only if it has a remainder of 3 when divided by 4. For example, 23 is awkward since

23 = 5 \times 4 + 3

.
A (true) theorem due to Fermat states that:
A positive integer is a squaresum if and only if each of its awkward prime factors occurs to an even power in its prime factorisation.
It follows that

5 \times 23^2

is a squaresum, since 23 occurs to the power 2, but

5 \times 23^3

is not, since 23 occurs to the power 3.
Which one of the following statements is not true?

A:Every square number is a squaresum.

B:If

N

and

M

are squaresums, then so is

NM

C:If

NM

is a squaresum, then

N

and

M

are squaresums.

D:If

N

is not a squaresum, then

kN

is a squaresum for some number

k

which is a product of awkward primes.

Practice Questions

Practice Question 1

The radius of an iron-56 atom is

3.0 \times 10^4

times greater than the radius of an iron-56 nucleus.

What is the value of

\frac{\text{density of an iron atom}}{\text{density of an iron nucleus}}

(3.0 \times 10^4)^{-3}

(3.0 \times 10^4)^{-2}

(3.0 \times 10^4)^{-1}

(3.0 \times 10^4)^1

(3.0 \times 10^4)^2

(3.0 \times 10^4)^3

Frequently asked questions

Is the square root of 16 both 4 and -4?

Technically, in the context of the TMUA and the symbol $\sqrt{}$ , the square root is defined as the positive result, which is $4$ . However, the number $16$ has two roots: the square root ( $4$ ) and the negative square root ( $-4$ ).

Can a negative number have a square root?

Within the scope of real numbers used in the TMUA, you cannot take the square root of a negative number because no real number multiplied by itself results in a negative value.

Why is the cube root of -8 equal to -2?

The cube root of $-8$ is $-2$ because $(-2) \times (-2) \times (-2) = -8$ . Since there are three factors of the negative number, the final product remains negative.

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