What is Square root? — Mathematics

Definition 65.1 Primary & Middle School Mathematics · Chapter 65 — Square Roots

Let $a \geq 0$ . The square root of $a$ , written $\sqrt a$ , is the unique nonnegative number whose square is $a$ :

\sqrt a \geq 0 \qquad\text{and}\qquad \left(\sqrt a\right)^2 = a .

Negative numbers have no square root, since every square is nonnegative.

Examples

Example 65.9 (Adding square roots)

Sums of square roots simplify only when the roots are alike. Compute $\sqrt{12} + \sqrt{27} - \sqrt{48}$ , simplifying each term first:

\begin{align*} \sqrt{12} &= \sqrt{4 \times 3} = 2\sqrt3, \\ \sqrt{27} &= \sqrt{9 \times 3} = 3\sqrt3, \\ \sqrt{48} &= \sqrt{16 \times 3} = 4\sqrt3, \end{align*}

so the sum is $2\sqrt3 + 3\sqrt3 - 4\sqrt3 = (2 + 3 - 4)\sqrt3 = \sqrt3$ .

Example 65.10 (Expanding with square roots)

The identities of algebra apply to square roots. Expand $\left(\sqrt5 + 2\right)^2$ with $(a+b)^2 = a^2 + 2ab + b^2$ :

\left(\sqrt5 + 2\right)^2 = 5 + 2 \times 2\sqrt5 + 4 = 9 + 4\sqrt5 .

And with the third identity, $(a+b)(a-b) = a^2 - b^2$ :

\left(\sqrt7 + \sqrt3\right)\left(\sqrt7 - \sqrt3\right) = 7 - 3 = 4 :

the product of two irrational numbers can be an integer.