In Exercises 21–32, simplify by factoring. __ √28

Verified step by step guidance

1. insert step 1> Start by expressing the number under the square root, 28, as a product of its prime factors.
2. insert step 2> The prime factorization of 28 is 2 \times 2 \times 7, or 2^2 \times 7.
3. insert step 3> Use the property of square roots that \sqrt{a \times b} = \sqrt{a} \times \sqrt{b} to separate the factors under the square root.
4. insert step 4> Apply the square root to the perfect square factor: \sqrt{2^2 \times 7} = \sqrt{2^2} \times \sqrt{7}.
5. insert step 5> Simplify \sqrt{2^2} to 2, resulting in 2 \times \sqrt{7}.

Key Concepts

Factoring

Factoring is the process of breaking down an expression into a product of simpler factors. In algebra, this often involves identifying common factors or applying techniques such as grouping, using the distributive property, or recognizing special products. Understanding how to factor is essential for simplifying expressions and solving equations.

Square Roots

A square root of a number is a value that, when multiplied by itself, gives the original number. For example, the square root of 28 can be expressed in terms of its prime factors. Recognizing how to simplify square roots by factoring out perfect squares is crucial for simplifying radical expressions.

Prime Factorization

Prime factorization is the process of expressing a number as the product of its prime factors. For instance, the prime factorization of 28 is 2^2 × 7. This concept is important in simplifying square roots, as it allows us to identify and extract perfect squares from under the radical sign, facilitating the simplification process.