Simplify the radical expressions in Exercises 58 - 62. ∛81

Verified step by step guidance

1. Identify the expression: \( \sqrt[3]{81} \). This is a cube root problem.
2. Recognize that 81 can be expressed as a power of a smaller number. Notice that 81 is \( 3^4 \) because \( 3 \times 3 \times 3 \times 3 = 81 \).
3. Rewrite the expression using the power of 3: \( \sqrt[3]{3^4} \).
4. Apply the property of radicals: \( \sqrt[n]{a^m} = a^{m/n} \). In this case, \( \sqrt[3]{3^4} = 3^{4/3} \).
5. Simplify the expression \( 3^{4/3} \) by separating it into \( 3^{3/3} \times 3^{1/3} \), which simplifies to \( 3 \times \sqrt[3]{3} \).

Key Concepts

Radical Expressions

Radical expressions involve roots, such as square roots or cube roots. They are written using the radical symbol (√) or the cube root symbol (∛). Understanding how to manipulate these expressions is essential for simplification, which often involves identifying perfect squares or cubes within the radicand.

Cube Roots

The cube root of a number x, denoted as ∛x, is a value that, when multiplied by itself three times, gives x. For example, ∛27 = 3 because 3 × 3 × 3 = 27. Recognizing perfect cubes, such as 1, 8, 27, and 64, helps in simplifying cube root expressions effectively.

Simplification Techniques

Simplification techniques for radical expressions include factoring the radicand into its prime factors and identifying any perfect squares or cubes. This process allows for the extraction of whole numbers from the radical, making the expression easier to work with. For instance, ∛81 can be simplified by recognizing that 81 = 27 × 3, leading to ∛81 = ∛(27 × 3) = ∛27 × ∛3 = 3∛3.