Question

which expression is equivalent to $2\sqrt3{x^2} \cdot \sqrt{16x}$, if $x > 0$?

a. $8x\sqrt6{x}$

b. $8\sqrt3{x}$

c. $8x$

d. $2x\sqrt3{16x}$

Explanation:

Step1: Simplify \(\sqrt{16x}\)

\(\sqrt{16x} = \sqrt{16}\cdot\sqrt{x}=4\sqrt{x}\), since \(\sqrt{16} = 4\) and \(x>0\).

Step2: Rewrite radicals as exponents

Recall that \(\sqrt[n]{a^m}=a^{\frac{m}{n}}\). So, \(\sqrt[3]{x^2}=x^{\frac{2}{3}}\) and \(\sqrt{x}=x^{\frac{1}{2}}\).
The original expression \(2\sqrt[3]{x^2}\cdot\sqrt{16x}\) becomes \(2x^{\frac{2}{3}}\cdot4x^{\frac{1}{2}}\).

Step3: Multiply coefficients and add exponents

Multiply the coefficients: \(2\times4 = 8\).
Add the exponents of \(x\) using the rule \(a^m\cdot a^n=a^{m + n}\): \(\frac{2}{3}+\frac{1}{2}=\frac{4 + 3}{6}=\frac{7}{6}\). So we have \(8x^{\frac{7}{6}}\).

Step4: Rewrite \(x^{\frac{7}{6}}\) as a radical

\(x^{\frac{7}{6}}=x^{1+\frac{1}{6}}=x\cdot x^{\frac{1}{6}}=x\sqrt[6]{x}\).
Thus, \(8x^{\frac{7}{6}} = 8x\sqrt[6]{x}\).

Answer:

A. \(8x\sqrt[6]{x}\)