Simplify each radical. See Example 5.3√27

Verified step by step guidance

1. Identify the expression: \(3\sqrt{27}\).
2. Recognize that \(27\) can be expressed as \(3^3\).
3. Rewrite the expression as \(3\sqrt{3^3}\).
4. Apply the property of radicals: \(\sqrt{a^n} = a^{n/2}\).
5. Simplify the expression by taking the square root of \(3^3\).

Key Concepts

Radical Expressions

Radical expressions involve roots, such as square roots or cube roots. The expression 3√27 indicates the cube root of 27, which asks for a number that, when multiplied by itself three times, equals 27. Understanding how to simplify these expressions is crucial for solving problems involving radicals.

Properties of Exponents

The properties of exponents are essential for simplifying radical expressions. For instance, the cube root can be expressed using fractional exponents, where 3√x is equivalent to x^(1/3). This understanding allows for easier manipulation and simplification of radical terms.

Simplifying Radicals

Simplifying radicals involves breaking down the expression into its simplest form. For example, the cube root of 27 simplifies to 3, since 3 × 3 × 3 = 27. Recognizing perfect cubes and applying simplification techniques is key to effectively solving radical expressions.