Simplify each radical. See Example 5. 3√27

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Identify the expression to simplify: \(3\sqrt{27}\). Here, the radical is \(\sqrt{27}\), which means the square root of 27, and it is multiplied by 3.

Factor the number inside the square root to find perfect squares. Since \(27 = 9 \times 3\), and 9 is a perfect square, rewrite the radical as \(\sqrt{9 \times 3}\).

Use the property of square roots that \(\sqrt{a \times b} = \sqrt{a} \times \sqrt{b}\) to separate the radical: \(\sqrt{9} \times \sqrt{3}\).

Simplify the square root of the perfect square: \(\sqrt{9} = 3\), so the expression inside the radical becomes \(3 \times \sqrt{3}\).

Multiply the simplified radical by the coefficient outside: \(3 \times 3 \times \sqrt{3} = 9 \sqrt{3}\). This is the simplified form of the original expression.

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Cube Roots

The cube root of a number is a value that, when multiplied by itself three times, gives the original number. For example, the cube root of 27 is 3 because 3 × 3 × 3 = 27. Understanding cube roots helps simplify expressions involving radicals with an index of 3.

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Complex Roots

Radical Simplification

Simplifying radicals involves rewriting the expression in its simplest form by factoring the radicand into perfect powers matching the root's index. For cube roots, this means extracting perfect cubes from under the radical to simplify the expression.

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Example 6

Properties of Exponents and Roots

Roots and exponents are inverse operations; for example, the cube root of a number is equivalent to raising that number to the power of 1/3. Using these properties allows conversion between radical and exponential forms, facilitating simplification.

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