Simplify each radical. See Example 5.- √160

Verified step by step guidance

1. insert step 1> Identify the prime factors of the number under the square root. For 160, start by dividing by the smallest prime number, 2.
2. insert step 2> Continue dividing by 2 until you can no longer do so evenly. 160 ÷ 2 = 80, 80 ÷ 2 = 40, 40 ÷ 2 = 20, 20 ÷ 2 = 10, 10 ÷ 2 = 5.
3. insert step 3> Now, express 160 as a product of its prime factors: 160 = 2^5 × 5.
4. insert step 4> Use the property of square roots that states √(a × b) = √a × √b. Apply this to the prime factorization: √160 = √(2^5 × 5).
5. insert step 5> Simplify the expression by taking out pairs of prime factors from under the square root. For every pair of 2s, one 2 comes out of the square root.

Key Concepts

Radical Simplification

Radical simplification involves rewriting a radical expression in its simplest form. This typically means factoring out perfect squares from under the radical sign. For example, √160 can be simplified by recognizing that 160 = 16 × 10, where 16 is a perfect square, allowing us to express √160 as √(16 × 10) = √16 × √10 = 4√10.

Perfect Squares

Perfect squares are numbers that can be expressed as the square of an integer. Common perfect squares include 1, 4, 9, 16, 25, and so on. Identifying perfect squares within a radical expression is crucial for simplification, as they can be factored out, making the radical easier to work with and understand.

Properties of Radicals

The properties of radicals include rules that govern how to manipulate radical expressions. Key properties include the product property (√a × √b = √(a × b)) and the quotient property (√a / √b = √(a / b)). Understanding these properties is essential for simplifying radicals effectively and performing operations involving them.