Question

which of the following are like radicals? choose four correct answers.
$2\sqrt{x^3y}$
$-2x\sqrt{xy^2}$
$-x\sqrt{x^2y^3}$
$x\sqrt{yx^3}$
$3x\sqrt{x^3y}$
$-12x\sqrt{x^3y}$

Explanation:

Step1: Simplify each radical

First, recall that like radicals have the same radicand (the expression inside the square root) when simplified. Let's simplify each radical:

• For \( 2\sqrt{x^3y} \):

Simplify \( \sqrt{x^3y} \). We can factor \( x^3 \) as \( x^2 \cdot x \), so \( \sqrt{x^3y}=\sqrt{x^2\cdot x\cdot y}=x\sqrt{xy} \) (assuming \( x\geq0 \)). Thus, \( 2\sqrt{x^3y} = 2x\sqrt{xy} \).

• For \( -2x\sqrt{xy^2} \):

Simplify \( \sqrt{xy^2} \). Factor \( y^2 \) out, so \( \sqrt{xy^2}=y\sqrt{x} \) (assuming \( y\geq0 \)). Thus, \( -2x\sqrt{xy^2}=-2xy\sqrt{x} \).

• For \( -x\sqrt{x^2y^3} \):

Simplify \( \sqrt{x^2y^3} \). Factor \( x^2 \) and \( y^2 \) out: \( \sqrt{x^2y^3}=xy\sqrt{y} \) (assuming \( x\geq0, y\geq0 \)). Thus, \( -x\sqrt{x^2y^3}=-x\cdot xy\sqrt{y}=-x^2y\sqrt{y} \).

• For \( x\sqrt{yx^3} \):

Simplify \( \sqrt{yx^3} \). Factor \( x^2 \) out: \( \sqrt{yx^3}=x\sqrt{xy} \) (assuming \( x\geq0 \)). Thus, \( x\sqrt{yx^3}=x\cdot x\sqrt{xy}=x^2\sqrt{xy} \).

• For \( 3x\sqrt{x^3y} \):

Simplify \( \sqrt{x^3y} \) as \( x\sqrt{xy} \) (assuming \( x\geq0 \)). Thus, \( 3x\sqrt{x^3y}=3x\cdot x\sqrt{xy}=3x^2\sqrt{xy} \).

• For \( -12x\sqrt{x^3y} \):

Simplify \( \sqrt{x^3y} \) as \( x\sqrt{xy} \) (assuming \( x\geq0 \)). Thus, \( -12x\sqrt{x^3y}=-12x\cdot x\sqrt{xy}=-12x^2\sqrt{xy} \).

Step2: Identify like radicals

Now, let's check the simplified forms:

• \( 2\sqrt{x^3y} = 2x\sqrt{xy} \)
• \( x\sqrt{yx^3} = x^2\sqrt{xy} \)
• \( 3x\sqrt{x^3y} = 3x^2\sqrt{xy} \)
• \( -12x\sqrt{x^3y} = -12x^2\sqrt{xy} \)

These four have the same radicand \( \sqrt{xy} \) (after simplification, the radicand part is \( xy \) inside the square root). Let's verify each:

1. \( 2\sqrt{x^3y} \): Simplifies to \( 2x\sqrt{xy} \)
2. \( x\sqrt{yx^3} \): Simplifies to \( x^2\sqrt{xy} \)
3. \( 3x\sqrt{x^3y} \): Simplifies to \( 3x^2\sqrt{xy} \)
4. \( -12x\sqrt{x^3y} \): Simplifies to \( -12x^2\sqrt{xy} \)

The other two: \( -2x\sqrt{xy^2} \) has radicand \( \sqrt{x} \) (simplified to \( -2xy\sqrt{x} \)) and \( -x\sqrt{x^2y^3} \) has radicand \( \sqrt{y} \) (simplified to \( -x^2y\sqrt{y} \)), so they are not like radicals with the first four.

Answer:

The four like radicals are:

• \( 2\sqrt{x^3y} \)
• \( x\sqrt{yx^3} \)
• \( 3x\sqrt{x^3y} \)
• \( -12x\sqrt{x^3y} \)