Number - Square Roots and Cube Roots

Square Roots and Area: The square root of a number $n$, written as $\sqrt{n}$, represents the side length of a square whose area is exactly $n$ units squared. Visually, if you have a square grid containing $16$ small squares, the length of one side is $\sqrt{16} = 4$ units.

Perfect Squares: These are integers that are the result of squaring another integer. Common examples include $1, 4, 9, 16, 25, 36, 49, 64, 81,$ and $100$. On a number line, perfect squares appear at increasing intervals as the base integer grows.

Cube Roots and Volume: The cube root of a number $n$, denoted as $\sqrt[3]{n}$, is the edge length of a 3D cube with a volume of $n$. For example, a rubik's cube made of $27$ smaller individual cubes has an edge length of $\sqrt[3]{27} = 3$ units.

Radical Notation: The symbol $\sqrt{}$ is called a radical. The number inside is the radicand, and the small number above the checkmark is the index. If no index is shown, it is assumed to be a square root (index $2$). For cube roots, the index is always $3$.

Negative Radicands: A square root of a negative number (e.g., $\sqrt{-16}$) is not a real number because no real number multiplied by itself results in a negative value. However, cube roots of negative numbers are real; for example, $\sqrt[3]{-64} = -4$ because $(-4) \times (-4) \times (-4) = -64$.

Estimation of Roots: For non-perfect squares like $\sqrt{10}$, we estimate the value by finding the two closest perfect squares. Since $\sqrt{9} < \sqrt{10} < \sqrt{16}$, we know that $3 < \sqrt{10} < 4$. On a number line, $\sqrt{10}$ would be positioned very close to $3$.

Inverse Operations: Squaring a number and taking its square root are inverse operations. Similarly, cubing and taking a cube root are inverses. For example, $5^2 = 25$ and $\sqrt{25} = 5$; $4^3 = 64$ and $\sqrt[3]{64} = 4$.