Question

what is \frac{\sqrt{12x^8}}{\sqrt{3x^2}} in simplest form, where x \ge 0?

• 2\sqrt{3}x^4
• \sqrt{15}x^5
• 2x^3
• 2x^2

Explanation:

Apply the quotient property of radicals

Using the Quotient Property of Radicals knowledge point

$$ \frac{\sqrt{12x^8}}{\sqrt{3x^2}} = \sqrt{\frac{12x^8}{3x^2}} $$

Simplify the fraction inside the radical

Using the Exponent Rules knowledge point

$$ \frac{12x^8}{3x^2} = 4x^{8-2} = 4x^6 $$

Take the square root of the simplified expression

Using the Radical Simplification knowledge point

$$ \sqrt{4x^6} = \sqrt{4} \cdot \sqrt{x^6} = 2x^3 \quad (\text{since } x \ge 0) $$

Answer:

• (A) \(2\sqrt{3}x^4\)
• (B) \(\sqrt{15}x^5\)
• (C) \(2x^3\) (Correct answer)
• (D) \(2x^2\)