Answer :
To determine which of the given expressions are like radicals, we need to simplify them and compare their radicands (the expressions under the square root) and any coefficients outside the roots.
Let's simplify each given expression step by step:
1. Expression: [tex]\(3 x \sqrt{x^2 y}\)[/tex]
Simplification:
[tex]\[ 3 x \sqrt{x^2 y} = 3 x \cdot \sqrt{x^2 y} = 3 x \cdot x \sqrt{y} = 3 x^2 \sqrt{y} \][/tex]
2. Expression: [tex]\(-12 x \sqrt{x^2 y}\)[/tex]
Simplification:
[tex]\[ -12 x \sqrt{x^2 y} = -12 x \cdot \sqrt{x^2 y} = -12 x \cdot x \sqrt{y} = -12 x^2 \sqrt{y} \][/tex]
3. Expression: [tex]\(-2 x \sqrt{x y^2}\)[/tex]
Simplification:
[tex]\[ -2 x \sqrt{x y^2} = -2 x \cdot \sqrt{x y^2} = -2 x \cdot y \sqrt{x} = -2 x y \sqrt{y} \][/tex]
4. Expression: [tex]\(x \sqrt{y x^2}\)[/tex]
Simplification:
[tex]\[ x \sqrt{y x^2} = x \cdot \sqrt{y x^2} = x \cdot x \sqrt{y} = x^2 \sqrt{y} \][/tex]
5. Expression: [tex]\(-x \sqrt{x^2 y^2}\)[/tex]
Simplification:
[tex]\[ -x \sqrt{x^2 y^2} = -x \cdot \sqrt{x^2 y^2} = -x \cdot x y = -x^2 y \][/tex]
6. Expression: [tex]\(2 \sqrt{x^2 y}\)[/tex]
Simplification:
[tex]\[ 2 \sqrt{x^2 y} = 2 \cdot \sqrt{x^2 y} = 2 x \sqrt{y} \][/tex]
Now, let's compare the simplified forms to see which expressions share the same radicand values and coefficients:
1. [tex]\(3 x^2 \sqrt{y}\)[/tex]
2. [tex]\(-12 x^2 \sqrt{y}\)[/tex]
4. [tex]\(x^2 \sqrt{y}\)[/tex]
We can see that expressions 1, 2, and 4 are like radicals because they have the same [tex]\(x^2 \sqrt{y}\)[/tex] term.
Expression 6 simplifies to [tex]\(2 x \sqrt{y}\)[/tex], which matches in general form but has a different coefficient and thus does not fit perfectly.
Hence, the pairs of like radicals are:
- [tex]\(3 x \sqrt{x^2 y}\)[/tex]
- [tex]\(-12 x \sqrt{x^2 y}\)[/tex]
- [tex]\(x \sqrt{y x^2}\)[/tex]
Therefore, checkboxes 1, 2, and 4 should be selected as the like radicals.
Let's simplify each given expression step by step:
1. Expression: [tex]\(3 x \sqrt{x^2 y}\)[/tex]
Simplification:
[tex]\[ 3 x \sqrt{x^2 y} = 3 x \cdot \sqrt{x^2 y} = 3 x \cdot x \sqrt{y} = 3 x^2 \sqrt{y} \][/tex]
2. Expression: [tex]\(-12 x \sqrt{x^2 y}\)[/tex]
Simplification:
[tex]\[ -12 x \sqrt{x^2 y} = -12 x \cdot \sqrt{x^2 y} = -12 x \cdot x \sqrt{y} = -12 x^2 \sqrt{y} \][/tex]
3. Expression: [tex]\(-2 x \sqrt{x y^2}\)[/tex]
Simplification:
[tex]\[ -2 x \sqrt{x y^2} = -2 x \cdot \sqrt{x y^2} = -2 x \cdot y \sqrt{x} = -2 x y \sqrt{y} \][/tex]
4. Expression: [tex]\(x \sqrt{y x^2}\)[/tex]
Simplification:
[tex]\[ x \sqrt{y x^2} = x \cdot \sqrt{y x^2} = x \cdot x \sqrt{y} = x^2 \sqrt{y} \][/tex]
5. Expression: [tex]\(-x \sqrt{x^2 y^2}\)[/tex]
Simplification:
[tex]\[ -x \sqrt{x^2 y^2} = -x \cdot \sqrt{x^2 y^2} = -x \cdot x y = -x^2 y \][/tex]
6. Expression: [tex]\(2 \sqrt{x^2 y}\)[/tex]
Simplification:
[tex]\[ 2 \sqrt{x^2 y} = 2 \cdot \sqrt{x^2 y} = 2 x \sqrt{y} \][/tex]
Now, let's compare the simplified forms to see which expressions share the same radicand values and coefficients:
1. [tex]\(3 x^2 \sqrt{y}\)[/tex]
2. [tex]\(-12 x^2 \sqrt{y}\)[/tex]
4. [tex]\(x^2 \sqrt{y}\)[/tex]
We can see that expressions 1, 2, and 4 are like radicals because they have the same [tex]\(x^2 \sqrt{y}\)[/tex] term.
Expression 6 simplifies to [tex]\(2 x \sqrt{y}\)[/tex], which matches in general form but has a different coefficient and thus does not fit perfectly.
Hence, the pairs of like radicals are:
- [tex]\(3 x \sqrt{x^2 y}\)[/tex]
- [tex]\(-12 x \sqrt{x^2 y}\)[/tex]
- [tex]\(x \sqrt{y x^2}\)[/tex]
Therefore, checkboxes 1, 2, and 4 should be selected as the like radicals.
