In Exercises 11–28, add or subtract as indicated. You will need to simplify terms to identify the like radicals.______ ___√4x - 12 + √x-3

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1. Identify the like radicals in the expression \( \sqrt{4x - 12} + \sqrt{x - 3} \).
2. Simplify each radical expression if possible. For \( \sqrt{4x - 12} \), factor out any perfect squares.
3. Rewrite \( \sqrt{4x - 12} \) as \( \sqrt{4(x - 3)} \) and simplify to \( 2\sqrt{x - 3} \).
4. Now the expression becomes \( 2\sqrt{x - 3} + \sqrt{x - 3} \).
5. Combine the like radicals: \( 2\sqrt{x - 3} + 1\sqrt{x - 3} = 3\sqrt{x - 3} \).

Key Concepts

Radicals

Radicals are expressions that involve roots, such as square roots, cube roots, etc. In this context, the square root symbol (√) indicates that we are dealing with the principal square root of a number or expression. Understanding how to manipulate radicals, including simplifying them and combining like terms, is essential for solving problems involving radical expressions.

Like Radicals

Like radicals are terms that have the same radicand (the number or expression inside the radical) and the same index. For example, √4 and √16 are not like radicals because their radicands differ. Identifying and combining like radicals is crucial when adding or subtracting radical expressions, as it allows for simplification and clearer results.

Simplification of Radicals

Simplification of radicals involves rewriting a radical expression in its simplest form. This can include factoring out perfect squares from under the radical sign or reducing the expression to eliminate any unnecessary radicals. Mastering this concept is vital for effectively performing operations such as addition and subtraction with radical expressions, as it ensures that the final answer is presented in the most concise manner.