Question

select the correct answer.

what is this expression in simplified form?

\\(\sqrt3{-864}\\)

\\(\bigcirc\\) \\(-4\sqrt3{6}\\)
\\(\bigcirc\\) \\(-6\sqrt3{4}\\)
\\(\bigcirc\\) \\(-12\sqrt3{6}\\)
\\(\bigcirc\\) \\(-6\sqrt3{12}\\)

Explanation:

⚡ Using what you learned: simplifying radical expressions

Step 1: Extract the negative sign

Since we are taking a cube root (an odd root) of a negative number, we can pull the negative sign out of the radical:

$$ \sqrt[3]{-864} = -\sqrt[3]{864} $$

Step 2: Find the prime factorization of 864

$$ 864 = 2 \times 432 $$

$$ 864 = 2^2 \times 216 $$

$$ 864 = 2^3 \times 108 $$

$$ 864 = 2^4 \times 54 $$

$$ 864 = 2^5 \times 27 $$

$$ 864 = 2^5 \times 3^3 $$

Group the factors into perfect cubes:

$$ 864 = 2^3 \times 3^3 \times 2^2 $$

$$ 864 = 8 \times 27 \times 4 $$

$$ 864 = 216 \times 4 $$

Step 3: Simplify the radical

$$ -\sqrt[3]{864} = -\sqrt[3]{216 \times 4} $$

$$ -\sqrt[3]{864} = -\sqrt[3]{216} \times \sqrt[3]{4} $$

$$ -\sqrt[3]{864} = -6\sqrt[3]{4} $$

Answer:

$$ -6\sqrt[3]{4} $$