Question

which of the following is equivalent to $2xsqrt3{x^2} - 3x^2sqrt{x} + xsqrt{x^3} - sqrt3{x^5}$?
options:

• $-2x^2sqrt{x} + xsqrt3{x^2}$
• $2xsqrt9{x^2}$
• $3x^2sqrt{x} - 4xsqrt3{x^2}$
• $-x^2sqrt{x}$

Explanation:

Step1: Rewrite radicals as exponents

Recall that $\sqrt[n]{x^m}=x^{\frac{m}{n}}$ and $x^a \cdot x^b = x^{a+b}$:

• $2x\sqrt[3]{x^2}=2x \cdot x^{\frac{2}{3}}=2x^{1+\frac{2}{3}}=2x^{\frac{5}{3}}$
• $-3x^2\sqrt{x}=-3x^2 \cdot x^{\frac{1}{2}}=-3x^{2+\frac{1}{2}}=-3x^{\frac{5}{2}}$
• $x\sqrt{x^3}=x \cdot x^{\frac{3}{2}}=x^{1+\frac{3}{2}}=x^{\frac{5}{2}}$
• $-\sqrt[3]{x^5}=-x^{\frac{5}{3}}$

Step2: Combine like terms

Group terms with the same exponent:

• For $x^{\frac{5}{3}}$: $2x^{\frac{5}{3}} - x^{\frac{5}{3}} = x^{\frac{5}{3}}$
• For $x^{\frac{5}{2}}$: $-3x^{\frac{5}{2}} + x^{\frac{5}{2}} = -2x^{\frac{5}{2}}$

Step3: Rewrite exponents back to radicals

Convert back to radical form:

• $x^{\frac{5}{3}}=x^{1+\frac{2}{3}}=x\sqrt[3]{x^2}$
• $-2x^{\frac{5}{2}}=-2x^{2+\frac{1}{2}}=-2x^2\sqrt{x}$

Combine the two results: $-2x^2\sqrt{x} + x\sqrt[3]{x^2}$

Answer:

$-2x^2\sqrt{x} + x\sqrt[3]{x^2}$ (the option in the green box)