Question

simplify the radical using the product property. assume that all variables represent real numbers.

\\\sqrt3{-48x^{12}}\\

\\\sqrt3{-48x^{12}} = \quad\\
(simplify your answer. type an exact answer, using radicals as needed.)

Explanation:

⚡ Using what you learned: Roots and Radicals

Step 1: Factor the radicand into perfect cubes

Identify the perfect cube factors of the coefficient and the variable term inside the cube root:

$$ -48 = -8 \cdot 6 = (-2)^3 \cdot 6 $$

$$ x^{12} = (x^4)^3 $$

Step 2: Apply the product property of radicals

Group the perfect cubes together and separate them from the remaining factors:

$$ \sqrt[3]{-48x^{12}} = \sqrt[3]{(-8 \cdot x^{12}) \cdot 6} $$

$$ \sqrt[3]{-48x^{12}} = \sqrt[3]{-8x^{12}} \cdot \sqrt[3]{6} $$

Step 3: Simplify the perfect cube root

Evaluate the cube root of the perfect cube terms:

$$ \sqrt[3]{-8x^{12}} = -2x^4 $$

Combine this with the remaining radical:

$$ -2x^4 \sqrt[3]{6} $$

Answer:

$$ -2x^4 \sqrt[3]{6} $$