To determine which of the given radical expressions are like radicals, we first need to simplify each expression and then compare them. Like radicals have the same radicand (the part under the radical) and the same index.
Let’s rewrite and simplify each expression step-by-step.
### Expression 1: [tex]\( 3x \sqrt{x^2 y} \)[/tex]
[tex]\[ 3x \sqrt{x^2 y} \][/tex]
The radicand here is [tex]\( x^2 y \)[/tex].
### Expression 2: [tex]\( -12x \sqrt{x^2 y} \)[/tex]
[tex]\[ -12x \sqrt{x^2 y} \][/tex]
The radicand here is [tex]\( x^2 y \)[/tex].
### Expression 3: [tex]\( -2x \sqrt{x y^2} \)[/tex]
[tex]\[ -2x \sqrt{x y^2} \][/tex]
We can rewrite the radicand [tex]\( x y^2 \)[/tex] as [tex]\( (xy) y \)[/tex]:
[tex]\[ -2x \sqrt{x y^2} = -2x \sqrt{xy \cdot y} = -2x \sqrt{xy^2} \][/tex]
The radicand here is [tex]\( x y^2 \)[/tex].
### Expression 4: [tex]\( x \sqrt{y x^2} \)[/tex]
[tex]\[ x \sqrt{y x^2} \][/tex]
We can rewrite the radicand [tex]\( y x^2 \)[/tex] as [tex]\( (xy)x \)[/tex]:
[tex]\[ x \sqrt{y x^2} = x \sqrt{x^2 y} \][/tex]
The radicand here is [tex]\( x^2 y \)[/tex].
### Expression 5: [tex]\( -x \sqrt{x^2 y^2} \)[/tex]
[tex]\[ -x \sqrt{x^2 y^2} \][/tex]
We can simplify the radicand [tex]\( x^2 y^2 \)[/tex] as:
[tex]\[ -x \sqrt{x^2 y^2} = -x \sqrt{(xy)^2} = -x (xy) \][/tex]
The radicand here is [tex]\( x^2 y^2 \)[/tex].
### Expression 6: [tex]\( 2 \sqrt{x^2 y} \)[/tex]
[tex]\[ 2 \sqrt{x^2 y} \][/tex]
The radicand here is [tex]\( x^2 y \)[/tex].
To summarize the simplified radicands for comparison:
- [tex]\( 3x \sqrt{x^2 y} \)[/tex] ⟶ radicand: [tex]\( x^2 y \)[/tex]
- [tex]\( -12x \sqrt{x^2 y} \)[/tex] ⟶ radicand: [tex]\( x^2 y \)[/tex]
- [tex]\( -2x \sqrt{x y^2} \)[/tex] ⟶ radicand: [tex]\( xy^2 \)[/tex]
- [tex]\( x \sqrt{x^2 y} \)[/tex] ⟶ radicand: [tex]\( x^2 y \)[/tex]
- [tex]\( -x \sqrt{x^2 y^2} \)[/tex] ⟶ radicand: [tex]\( x^2 y^2 \)[/tex]
- [tex]\( 2 \sqrt{x^2 y} \)[/tex] ⟶ radicand: [tex]\( x^2 y \)[/tex]
Like radicals have the same radicand. Therefore:
Expressions with radicand [tex]\( x^2 y \)[/tex] are:
- [tex]\( 3x \sqrt{x^2 y} \)[/tex]
- [tex]\( -12x \sqrt{x^2 y} \)[/tex]
- [tex]\( x \sqrt{x^2 y} \)[/tex]
- [tex]\( 2 \sqrt{x^2 y} \)[/tex]
Therefore, the like radicals are:
- [tex]\( 3x \sqrt{x^2 y} \)[/tex]
- [tex]\( -12x \sqrt{x^2 y} \)[/tex]
- [tex]\( x \sqrt{x^2 y} \)[/tex]
- [tex]\( 2 \sqrt{x^2 y} \)[/tex]
These four expressions have the same radicand [tex]\( x^2 y \)[/tex] and are therefore like radicals.
