Lesson 4.3: Roots and Radicals

New Concept

This lesson introduces square roots, the inverse of squaring a number. You'll learn what the radical symbol $\sqrt{N}$ means and how to find or approximate square roots to solve equations and find missing side lengths in right triangles.

What’s next

You'll start by mastering the definition of a square root, then apply this concept through a series of interactive examples and practice problems.

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

An irrational number is one that cannot be written as a fraction. The square roots of whole numbers that are not perfect squares are irrational. We can find an approximation for these numbers by using a calculator and rounding to the desired number of decimal places. For example, $\sqrt{5} \approx 2.236067977...$.

Examples

• To approximate $\sqrt{11}$ to three decimal places, a calculator shows 3.3166247... which rounds to 3.317.
• To approximate $\sqrt{175}$ to three decimal places, a calculator shows 13.228756... which rounds to 13.229.
• To approximate $\sqrt{0.06}$ to three decimal places, a calculator shows 0.2449489... which rounds to 0.245.

Explanation

Not all numbers have a neat, tidy square root. For numbers like $\sqrt{2}$ or $\sqrt{7}$, the decimal value goes on forever without repeating. Since we cannot write the exact decimal, we use a calculator to find a close approximation.

Property

Squaring and taking square roots are opposite operations; each operation undoes the effects of the other. If you square a positive number and then take the square root of the result, you get back to the original number. To solve an equation where the variable is squared, such as $n^2 = 36$, we take the square root of both sides to find $n = \sqrt{36} = 6$.

Examples

• If a positive number $x$ satisfies $x^2 = 121$, we can find $x$ by taking the square root: $x = \sqrt{121} = 11$.
• To simplify $(\sqrt{31})^2$, the square and square root cancel each other out, leaving 31.
• To simplify $\sqrt{19^2}$, the square root undoes the square, and the result is 19.

Explanation

Think of squaring and taking a square root as a "do" and "undo" pair. If you square a number (like $8 \rightarrow 64$) and then immediately take the square root ($\sqrt{64} \rightarrow 8$), you always return to your starting number.

Property

In a right triangle, the side opposite the right angle is called the hypotenuse, and the other two sides are the legs. If the letter $c$ stands for the length of the hypotenuse, and the lengths of the two legs are denoted by $a$ and $b$, then $a^2 + b^2 = c^2$.

Examples

• A right triangle has legs of length 5 cm and 12 cm. The hypotenuse $c$ is found with $5^2 + 12^2 = c^2$, so $25 + 144 = c^2$, which means $c^2 = 169$ and $c = \sqrt{169} = 13$ cm.
• The hypotenuse of a right triangle is 17 inches long and one leg is 15 inches long. The other leg $a$ is found with $a^2 + 15^2 = 17^2$, so $a^2 + 225 = 289$, which means $a^2 = 64$ and $a = \sqrt{64} = 8$ inches.
• The hypotenuse of a right triangle is 9 meters and one leg is 5 meters. The other leg $b$ is found with $5^2 + b^2 = 9^2$, so $25 + b^2 = 81$, which means $b^2 = 56$ and $b = \sqrt{56} \approx 7.48$ meters.

Explanation

This famous theorem is a special formula just for right triangles. It creates a powerful link between the lengths of all three sides. If you know the lengths of any two sides, you can always find the length of the third.