Simplify each radical. Assume all variables represent positive real numbers. ∛(27 + a³)

Verified step by step guidance

1. Recognize that the expression inside the cube root is \$27 + a^{3}\(. Notice that \)27\( is a perfect cube since \)27 = 3^{3}$.
2. Rewrite the expression inside the cube root as a sum of cubes: \$27 + a^{3} = 3^{3} + a^{3}$.
3. Recall the sum of cubes factorization formula: \(x^{3} + y^{3} = (x + y)(x^{2} - xy + y^{2})\). Here, \(x = 3\) and \(y = a\).
4. Apply the sum of cubes formula to factor the expression inside the cube root: \$3^{3} + a^{3} = (3 + a)(3^{2} - 3a + a^{2}) = (3 + a)(9 - 3a + a^{2})$.
5. Use the property of radicals that \(\sqrt[3]{xy} = \sqrt[3]{x} \times \sqrt[3]{y}\) to write \(\sqrt[3]{(3 + a)(9 - 3a + a^{2})} = \sqrt[3]{3 + a} \times \sqrt[3]{9 - 3a + a^{2}}\). Since \$3 + a\( is inside the cube root and cannot be simplified further, the simplified form is \)\sqrt[3]{3 + a} \times \sqrt[3]{9 - 3a + a^{2}}$.

Key Concepts

Properties of Radicals

Radicals represent roots, such as square roots or cube roots. Simplifying radicals involves rewriting the expression to its simplest root form, often by factoring the radicand into perfect powers that match the root's degree.

Cube Roots and Perfect Cubes

The cube root of a number is a value that, when cubed, returns the original number. Recognizing perfect cubes like 27 (3³) helps simplify expressions under cube roots by extracting these perfect cubes outside the radical.

Sum of Cubes Factorization

The sum of cubes formula, a³ + b³ = (a + b)(a² - ab + b²), allows factoring expressions like 27 + a³. This factorization is useful for simplifying radicals involving sums of cubes by breaking them into products that may simplify further.