Lesson 3: Square Roots and Cube Roots

Property

A number $s$ is called a square root of $N$ if $s^2 = N$. We use a special symbol called a radical sign, $\sqrt{\hphantom{0}}$, to denote the positive square root of a number. For example, $\sqrt{16}$ means "the positive square root of 16," so $\sqrt{16} = 4$. Numbers such as 16 and 25 are called perfect squares because they are the squares of whole numbers.

Examples

• 4 is a square root of 16 because $4^2 = 16$.
• 9 is a square root of 81 because $9^2 = 81$.
• $\frac{3}{5}$ is a square root of $\frac{9}{25}$ because $(\frac{3}{5})^2 = \frac{9}{25}$.

Explanation

Think of a square root as the reverse of squaring a number. If you know the area of a square, the square root tells you the length of its side. It answers the question: "What number, when multiplied by itself, gives this result?"

Property

A cube root of a number $a$ is a number $b$ such that $b^3 = a$. The cube root is denoted by the symbol $\sqrt[3]{a}$.

Property

The square root of a perfect square is an integer. The cube root of a perfect cube is an integer.

• If $x = n^2$ for some integer $n$, then $\sqrt{x} = n$.
• If $y = m^3$ for some integer $m$, then $\sqrt[3]{y} = m$.

Examples

• The square root of the perfect square $81$ is an integer: $\sqrt{81} = \sqrt{9^2} = 9$.
• The cube root of the perfect cube $64$ is an integer: $\sqrt[3]{64} = \sqrt[3]{4^3} = 4$.
• The cube root of the negative perfect cube $-125$ is an integer: $\sqrt[3]{-125} = \sqrt[3]{(-5)^3} = -5$.

Explanation

A perfect square is the result of squaring an integer, and a perfect cube is the result of cubing an integer. When you take the square root of a perfect square, you are simply finding the original integer that was squared. Similarly, taking the cube root of a perfect cube reveals the original integer that was cubed. This is why the roots of perfect squares and perfect cubes are always integers.