Pythagorean Theorem and Simplifying Radicals

The Pythagorean Theorem states that in a right triangle, the square of the length of the hypotenuse (c) is equal to the sum of the squares of the lengths of the other two sides (a and b). It can be expressed as: $$a^2 + b^2 = c^2$$ .

To simplify the square root of a number, factor the number into its prime factors, then pair the factors. Each pair can be taken out of the square root as a single factor.

The simplest radical form of $$\frac{10}{\text{sqrt}{12}}$$ is $$\frac{5\text{sqrt}{3}}{3}$$ .

A radical expression is an expression that includes a root, such as a square root, cube root, etc. For example, $$\text{sqrt}{x}$$ is a radical expression.

$$\text{sqrt}{3600} = 60$$ .

Expressing an answer in simplest radical form means to simplify the radical expression as much as possible, removing any perfect squares from under the radical.

$$\text{sqrt}{125} = 5\text{sqrt}{5}$$ .

$$\text{sqrt}{a^2b} = a\text{sqrt}{b}$$ , where a is a non-negative number.

In a 30-60-90 triangle, the lengths of the sides are in the ratio 1:√3:2. The side opposite the 30° angle is the shortest.

$$2\text{sqrt}{64} = 16$$ .