Question

add or subtract, as indicated. assume all variables are greater than or equal to zero.
\\(- 5x\sqrt3{24x} + 8\sqrt3{3x^4}\\)

select the correct choice below and, if necessary, fill in the answer box to complete your choice.

\\(\bigcirc\\) a.
\\(\quad\\) (type an exact answer, using radicals as needed.)
\\(\bigcirc\\) b. the expression is fully simplified.

Explanation:

Step1: Simplify the first radical

Factor 24x inside the cube root: \(24x = 8\times3x\), so \(\sqrt[3]{24x}=\sqrt[3]{8\times3x}\). Since \(\sqrt[3]{8} = 2\), we have \(\sqrt[3]{24x}=2\sqrt[3]{3x}\). Then the first term \(-5x\sqrt[3]{24x}\) becomes \(-5x\times2\sqrt[3]{3x}=-10x\sqrt[3]{3x}\).

Step2: Simplify the second radical

Factor \(x^4\) inside the cube root: \(x^4 = x^3\times x\), so \(\sqrt[3]{3x^4}=\sqrt[3]{3x^3\times x}\). Since \(\sqrt[3]{x^3}=x\), we have \(\sqrt[3]{3x^4}=x\sqrt[3]{3x}\). Then the second term \(8\sqrt[3]{3x^4}\) becomes \(8\times x\sqrt[3]{3x}=8x\sqrt[3]{3x}\).

Step3: Combine like terms

Now we have \(-10x\sqrt[3]{3x}+8x\sqrt[3]{3x}\). Combining the coefficients: \((-10 + 8)x\sqrt[3]{3x}=-2x\sqrt[3]{3x}\).

Answer:

A. \(-2x\sqrt[3]{3x}\)