Question

select the correct answer.

what is this expression in simplified form?

\\(\sqrt3{108}\\)

\\(4 \sqrt3{3}\\)

\\(3 \sqrt3{2}\\)

\\(3 \sqrt3{4}\\)

\\(2 \sqrt3{9}\\)

Explanation:

⚡ Using what you learned: simplifying radical expressions

Step 1: Find perfect cube factors

To simplify a cube root, look for factors of the number under the radical that are perfect cubes (such as \(8, 27, 64, \dots\)).

$$ 108 = 27 \times 4 $$

Since \(27\) is a perfect cube (\(3^3 = 27\)), we can rewrite the expression:

$$ \sqrt[3]{108} = \sqrt[3]{27 \times 4} $$

Step 2: Simplify the radical

Apply the product property of radicals, \(\sqrt[3]{a \cdot b} = \sqrt[3]{a} \cdot \sqrt[3]{b}\):

$$ \sqrt[3]{27 \times 4} = \sqrt[3]{27} \cdot \sqrt[3]{4} $$

Since \(\sqrt[3]{27} = 3\):

$$ 3\sqrt[3]{4} $$

Answer:

\(3\sqrt[3]{4}\)