Perform the indicated operations. Assume all variables represent positive real numbers. 2∛16 + ∛54

검증된 단계별 안내

Rewrite each cube root expression in terms of prime factors or simpler components. For example, express \(\sqrt[3]{16}\) and \(\sqrt[3]{54}\) using their prime factorizations: \(16 = 2^4\) and \(54 = 2 \times 3^3\).

Use the property of cube roots that \(\sqrt[3]{a^b} = a^{\frac{b}{3}}\) to simplify each term. For instance, \(\sqrt[3]{2^4} = 2^{\frac{4}{3}}\) and \(\sqrt[3]{2 \times 3^3} = \sqrt[3]{2} \times \sqrt[3]{3^3} = \sqrt[3]{2} \times 3\).

Rewrite the original expression \(2 \sqrt[3]{16} + \sqrt[3]{54}\) using the simplified forms from step 2, so it becomes \(2 \times 2^{\frac{4}{3}} + 3 \times \sqrt[3]{2}\).

Factor out the common cube root term \(\sqrt[3]{2} = 2^{\frac{1}{3}}\) from both terms to combine them. This means expressing each term as a multiple of \(2^{\frac{1}{3}}\).

After factoring, combine the coefficients and write the expression as a single term involving \(\sqrt[3]{2}\). This completes the simplification of the original expression.

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주요 개념

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Radical Expressions and Cube Roots

A radical expression involves roots such as square roots or cube roots. The cube root of a number x, denoted ∛x, is the value that when cubed gives x. Understanding how to interpret and simplify cube roots is essential for manipulating these expressions.

Simplifying Radicals

Simplifying radicals involves factoring the number inside the root to extract perfect powers. For cube roots, this means expressing the radicand as a product of a perfect cube and another factor, allowing the cube root of the perfect cube to be taken outside the radical.

Adding Radical Expressions

Radical expressions can only be added or subtracted if they have the same radicand and index. After simplifying each radical, like terms can be combined by adding their coefficients. This concept is crucial to correctly perform the indicated operations.