Simplify each radical. Assume all variables represent positive real numbers. ∛250

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Recognize that the problem asks to simplify the cube root of 250, written as \(\sqrt[3]{250}\).

Factor 250 into its prime factors: \(250 = 2 \times 5^3\).

Rewrite the cube root using the prime factorization: \(\sqrt[3]{2 \times 5^3}\).

Use the property of radicals that \(\sqrt[3]{a \times b} = \sqrt[3]{a} \times \sqrt[3]{b}\) to separate the factors: \(\sqrt[3]{2} \times \sqrt[3]{5^3}\).

Simplify \(\sqrt[3]{5^3}\) to 5, since the cube root and the cube cancel each other out, leaving the expression as \(5 \times \sqrt[3]{2}\).

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Simplifying Radicals

Simplifying radicals involves expressing the radical in its simplest form by factoring out perfect powers. For cube roots, this means identifying factors that are perfect cubes and separating them from the radical to simplify the expression.

Prime Factorization

Prime factorization is breaking down a number into its prime factors. This helps in identifying perfect cubes within the number, which can be taken out of the cube root to simplify the radical.

Properties of Cube Roots

The cube root of a product equals the product of the cube roots: ∛(a·b) = ∛a · ∛b. This property allows us to separate the radicand into factors, simplify perfect cubes, and rewrite the expression in a simpler form.

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