Simplify the radical expression. ∛81

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Identify the expression to simplify: the cube root of 81, written as \(\sqrt[3]{81}\).

Express 81 as a product of its prime factors: \(81 = 3^4\).

Rewrite the cube root using the prime factorization: \(\sqrt[3]{3^4}\).

Apply the property of radicals that \(\sqrt[3]{a^b} = a^{\frac{b}{3}}\), so rewrite as \(3^{\frac{4}{3}}\).

Separate the exponent into an integer and fractional part: \(3^{1 + \frac{1}{3}} = 3^1 \times 3^{\frac{1}{3}}\), which simplifies to \(3 \times \sqrt[3]{3}\).

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주요 개념

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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, ∛27 = 3 because 3 × 3 × 3 = 27. Understanding cube roots helps simplify expressions involving radicals with an index of 3.

Prime Factorization

Prime factorization involves breaking down a number into its prime number factors. This method is useful for simplifying radicals by identifying perfect cubes within the factors, making it easier to extract cube roots.

Simplifying Radical Expressions

Simplifying radicals means rewriting them in their simplest form by extracting perfect powers from under the radical sign. For cube roots, this involves pulling out factors that are perfect cubes and reducing the expression accordingly.