Simplify each radical. See Example 5. - √160

Verified step by step guidance

1. Start by expressing the number inside the radical, 160, as a product of its prime factors. This helps identify perfect squares. For 160, find its prime factorization.
2. Write 160 as a product of prime factors: \(160 = 2 \times 2 \times 2 \times 2 \times 2 \times 5\) or \(160 = 2^5 \times 5\).
3. Group the prime factors into pairs because \(\sqrt{a^2} = a\). Here, group the factors of 2 into pairs: \(2^5 = (2^2) \times (2^2) \times 2\).
4. Rewrite the square root using these groups: \(\sqrt{160} = \sqrt{(2^2) \times (2^2) \times 2 \times 5}\).
5. Simplify by taking the square root of each perfect square out of the radical: \(\sqrt{(2^2)} = 2\), so the expression becomes \(2 \times 2 \times \sqrt{2 \times 5}\), which simplifies to \(4 \sqrt{10}\).

Key Concepts

Simplifying Radicals

Simplifying radicals involves expressing a square root in its simplest form by factoring the radicand into perfect squares and other factors. For example, √160 can be broken down into √(16 × 10), allowing the square root of 16 to be simplified to 4.

Prime Factorization

Prime factorization is the process of breaking down a number into its prime factors. This helps identify perfect square factors within the radicand, which can be taken out of the square root to simplify the expression.

Properties of Square Roots

The property √(a × b) = √a × √b allows the separation of a square root of a product into the product of square roots. This property is essential for simplifying radicals by isolating perfect squares from other factors.