Define the key term like radicals.
The correct answer and explanation is:
A radical is a mathematical expression that includes a root, most commonly a square root, cube root, or other roots. The term “radical” refers to the symbol used for roots, typically written as “√”. For example, the square root of 9 is represented as √9, which simplifies to 3. Radicals express numbers or quantities that cannot be simplified into a more straightforward form using basic arithmetic operations like addition, subtraction, multiplication, or division.
A key feature of radicals is the index, which indicates which root is being taken. For instance, the square root symbol “√” has an implicit index of 2, but for cube roots, a smaller number (like “∛”) is used. The expression “∛8” would represent the cube root of 8, which equals 2. Roots of higher degrees can be represented similarly, like the fourth root (⁴√), fifth root (⁵√), etc.
Radicals are useful for expressing values that cannot be neatly represented as rational numbers. For instance, the square root of 2 (√2) is an irrational number, meaning it cannot be expressed exactly as a fraction or decimal. In such cases, radicals provide a more compact way of representing these numbers.
Radical expressions can be simplified, added, subtracted, multiplied, and divided according to certain rules. For example, when multiplying two square roots like √2 * √3, the result would be √6. Similarly, when adding or subtracting square roots, you can only combine terms that have the same radicand (the number under the root). For example, √3 + 2√3 would equal 3√3, since both terms involve the same radicand of 3.
Understanding and working with radicals is a fundamental skill in algebra and higher mathematics, as it is used to solve equations involving square roots or other roots.