In Exercises 75–82, add or subtract terms whenever possible. √3+³√15

Verified step by step guidance

1. Identify the terms involved: $\sqrt{3}$ and $\sqrt[3]{15}$.
2. Recognize that $\sqrt{3}$ is a square root and $\sqrt[3]{15}$ is a cube root.
3. Understand that terms with different root indices (square root vs. cube root) cannot be directly added or subtracted.
4. Conclude that the expression $\sqrt{3}+\sqrt[3]{15}$ is already in its simplest form.
5. Note that no further simplification is possible unless additional context or constraints are provided.

Key Concepts

Radicals

Radicals are expressions that involve roots, such as square roots (√) and cube roots (³√). Understanding how to simplify and manipulate these expressions is crucial for combining them. For example, √3 is a square root, while ³√15 is a cube root, and they cannot be directly added or subtracted unless they can be simplified to a common form.

Like Terms

Like terms are terms that contain the same variable raised to the same power or, in the case of radicals, the same root. To add or subtract terms, it is essential to identify like terms. In the expression √3 + ³√15, these terms are not like terms, as they involve different roots, which means they cannot be combined directly.

Simplification of Radicals

Simplification of radicals involves rewriting a radical expression in its simplest form. This may include factoring out perfect squares or cubes from under the radical sign. For instance, if a radical can be simplified to a simpler radical or a whole number, it can help in combining terms. However, in this case, neither √3 nor ³√15 can be simplified further, so they remain separate.