In Exercises 11–28, add or subtract as indicated. You will need to simplify terms to identify the like radicals.___ ___5√45x - 2√20x

Verified step by step guidance

1. Identify the like radicals by simplifying each radical expression.
2. Simplify \( \sqrt{45x} \) by factoring 45 as \( 9 \times 5 \) and using \( \sqrt{9} = 3 \), so \( \sqrt{45x} = 3\sqrt{5x} \).
3. Simplify \( \sqrt{20x} \) by factoring 20 as \( 4 \times 5 \) and using \( \sqrt{4} = 2 \), so \( \sqrt{20x} = 2\sqrt{5x} \).
4. Rewrite the original expression using the simplified radicals: \( 5(3\sqrt{5x}) - 2(2\sqrt{5x}) \).
5. Combine the like terms by factoring out \( \sqrt{5x} \): \( (5 \times 3 - 2 \times 2)\sqrt{5x} \).

Key Concepts

Radical Expressions

Radical expressions involve roots, such as square roots, cube roots, etc. In this context, we are dealing with square roots, which are denoted by the radical symbol (√). Understanding how to manipulate these expressions, including simplifying them, is crucial for performing operations like addition and subtraction.

Like Radicals

Like radicals are terms that have the same radicand (the number or expression inside the radical) and the same index. For example, √20 and √45 are not like radicals because their radicands differ. To add or subtract radical expressions, it is essential to simplify them to like radicals, allowing for straightforward combination of the terms.

Simplifying Radicals

Simplifying radicals involves breaking down the radicand into its prime factors and extracting any perfect squares. For instance, √45 can be simplified to 3√5, as 45 = 9 × 5 and √9 = 3. This process is necessary to identify like radicals and perform addition or subtraction accurately.