To determine which of the given expressions are like radicals, we need to simplify each expression and see which ones reduce to the same form.
Let's simplify each expression step by step.
1. [tex]\( 3x \sqrt{x^2 y} \)[/tex]:
[tex]\[
3x \sqrt{x^2 y} = 3x \sqrt{x^2} \sqrt{y} = 3x \cdot x \sqrt{y} = 3x^2 \sqrt{y}
\][/tex]
2. [tex]\( -12x \sqrt{x^2 y} \)[/tex]:
[tex]\[
-12x \sqrt{x^2 y} = -12x \sqrt{x^2} \sqrt{y} = -12x \cdot x \sqrt{y} = -12x^2 \sqrt{y}
\][/tex]
3. [tex]\( -2x \sqrt{x y^2} \)[/tex]:
[tex]\[
-2x \sqrt{x y^2} = -2x \sqrt{x} \sqrt{y^2} = -2x \sqrt{x} \cdot y = -2xy \sqrt{x}
\][/tex]
4. [tex]\( x \sqrt{y x^2} \)[/tex]:
[tex]\[
x \sqrt{y x^2} = x \sqrt{y} \sqrt{x^2} = x \sqrt{y} \cdot x = x^2 \sqrt{y}
\][/tex]
5. [tex]\( -x \sqrt{x^2 y^2} \)[/tex]:
[tex]\[
-x \sqrt{x^2 y^2} = -x \sqrt{x^2} \sqrt{y^2} = -x \cdot x \cdot y = -xyx = -x^2 y
\][/tex]
6. [tex]\( 2 \sqrt{x^2 y} \)[/tex]:
[tex]\[
2 \sqrt{x^2 y} = 2 \sqrt{x^2} \sqrt{y} = 2 \cdot x \sqrt{y} = 2x \sqrt{y}
\][/tex]
Now, let's compare the simplified expressions:
1. [tex]\( 3x^2 \sqrt{y} \)[/tex]
2. [tex]\( -12x^2 \sqrt{y} \)[/tex]
3. [tex]\( -2xy \sqrt{x} \)[/tex]
4. [tex]\( x^2 \sqrt{y} \)[/tex]
5. [tex]\( -x^2 y \)[/tex]
6. [tex]\( 2x \sqrt{y} \)[/tex]
Expressions that have the same radical terms and coefficients with the same variable powers:
- [tex]\( 3x^2 \sqrt{y} \)[/tex]
- [tex]\( -12x^2 \sqrt{y} \)[/tex]
- [tex]\( x^2 \sqrt{y} \)[/tex]
These expressions all have the form [tex]\( kx^2 \sqrt{y} \)[/tex], where [tex]\( k \)[/tex] is a coefficient.
Thus, the expressions that are like radicals are:
- [tex]\( 3x \sqrt{x^2 y} \)[/tex]
- [tex]\( -12x \sqrt{x^2 y} \)[/tex]
- [tex]\( x \sqrt{y x^2} \)[/tex]
These correspond to the first, second, and fourth expressions from the original list. Therefore, check the boxes:
- [tex]\( 3x \sqrt{x^2 y} \)[/tex]
- [tex]\( -12x \sqrt{x^2 y} \)[/tex]
- [tex]\( x \sqrt{y x^2} \)[/tex]
