Lesson 1: Finding Square Roots

Property

The symbol $\sqrt{A}$ (square root) indicates a number $a$ whose square is $A$: $a^2 = A$. The square root $\sqrt{A}$ is only defined for non-negative numbers $A$. A positive integer whose square root is a positive integer is called a perfect square.

Examples

• Since $8^2 = 64$, the square root of 64 is 8. We write this as $\sqrt{64} = 8$. The number 64 is a perfect square.
• The number 50 is not a perfect square. Its square root, $\sqrt{50}$, is a number that, when multiplied by itself, equals 50.
• To find the number whose square is 121, we are looking for $\sqrt{121}$. Since $11 \times 11 = 121$, the answer is $11$.

Explanation

A square root is the opposite of squaring a number. If you know the area of a square, the square root tells you the side length. Perfect squares are special because their square roots are nice, neat whole numbers!

Property

The symbol $\sqrt{\ }$ is called a radical sign, and the number inside is called the radicand.
The positive square root of a number is called the principal square root.

Examples

• The principal square root of 81 is written as $\sqrt{81}$, which equals 9.
• To express the negative square root of 36, we write $-\sqrt{36}$, which equals $-6$.
• The expression $\pm\sqrt{100}$ represents both square roots of 100, which means 10 or $-10$.

Explanation

The radical symbol $\sqrt{\ }$ is a specific instruction to find only the positive square root, known as the principal root.

Property

When using the order of operations, treat the radical sign as a grouping symbol. Simplify any expressions under the radical sign before performing other operations. For any negative number, there is no real number solution for its square root.

Examples

• To simplify $\sqrt{9 + 16}$, you must first add the numbers inside the radical to get $\sqrt{25}$, which equals $5$.
• To simplify $\sqrt{9} + \sqrt{16}$, you find each square root separately before adding: $3 + 4 = 7$. Note this is different from the first example.
• The expression $\sqrt{-169}$ is not a real number because no real number multiplied by itself can result in $-169$.

Explanation

The radical sign acts like parentheses, so you must simplify the expression inside it first. You cannot take the square root of a negative number in the real number system because squaring any real number always results in a non-negative value.