What is how to simplify radicals?

Simplifying Radicals

Simplifying radicals involves expressing a radical in its simplest form. This means removing any perfect square factors from under the radical sign. Here's how to do it:

1. Understand Perfect Squares: Familiarize yourself with perfect squares. Perfect squares are numbers that result from squaring an integer (e.g., 4, 9, 16, 25, 36, 49, 64, 81, 100, etc.).

2. Factor the Radicand: Break down the number under the radical sign (the radicand) into its prime factors. Look for factors that are perfect squares.

3. Identify Perfect Square Factors: Identify any perfect square factors within the factored radicand.

4. Extract Perfect Squares: Take the square root of the perfect square factors and move them outside the radical. Remember that √(a*b) = √a * √b

5. Simplify: Multiply any terms outside the radical and leave the remaining factors inside the radical.

Example 1:

Simplify √32

• Factor 32: 32 = 2 * 2 * 2 * 2 * 2 = 2⁵
• Identify Perfect Square: √32 = √(16 * 2)
• Extract Perfect Square: √16 * √2 = 4√2

Example 2:

Simplify √75

• Factor 75: 75 = 3 * 5 * 5
• Identify Perfect Square: √75 = √(25 * 3)
• Extract Perfect Square: √25 * √3 = 5√3

Simplifying Radicals with Variables

When variables are involved, you can simplify radicals by using the following properties:

• √(x²) = |x| (Absolute value is important if you don't know the sign of x)
• √(xⁿ) = x^n/2 (If n is even and positive)

Example:

Simplify √(x³)

• Rewrite: √(x² * x)
• Extract: √(x²) * √x = x√x

Important Subjects

• Perfect Squares
• Radicand
• Prime Factorization
• Square Root