To determine which radicals are like after simplifying, let's simplify each of the given radical expressions step-by-step.
1. Simplify [tex]\(\sqrt{50 x^2}\)[/tex]:
- [tex]\(\sqrt{50 x^2} = \sqrt{50} \cdot \sqrt{x^2}\)[/tex]
- [tex]\(\sqrt{50}\)[/tex] can be simplified as [tex]\(\sqrt{25 \cdot 2} = \sqrt{25} \cdot \sqrt{2} = 5\sqrt{2}\)[/tex]
- [tex]\(\sqrt{x^2} = x\)[/tex]
- Thus, [tex]\(\sqrt{50 x^2} = 5\sqrt{2} \cdot x\)[/tex]
2. Simplify [tex]\(\sqrt{32 x}\)[/tex]:
- [tex]\(\sqrt{32 x} = \sqrt{32} \cdot \sqrt{x}\)[/tex]
- [tex]\(\sqrt{32}\)[/tex] can be simplified as [tex]\(\sqrt{16 \cdot 2} = \sqrt{16} \cdot \sqrt{2} = 4\sqrt{2}\)[/tex]
- Thus, [tex]\(\sqrt{32 x} = 4\sqrt{2} \cdot \sqrt{x}\)[/tex]
3. Simplify [tex]\(\sqrt{18 n}\)[/tex]:
- [tex]\(\sqrt{18 n} = \sqrt{18} \cdot \sqrt{n}\)[/tex]
- [tex]\(\sqrt{18}\)[/tex] can be simplified as [tex]\(\sqrt{9 \cdot 2} = \sqrt{9} \cdot \sqrt{2} = 3\sqrt{2}\)[/tex]
- Thus, [tex]\(\sqrt{18 n} = 3\sqrt{2} \cdot \sqrt{n}\)[/tex]
4. Simplify [tex]\(\sqrt{72 x^2}\)[/tex]:
- [tex]\(\sqrt{72 x^2} = \sqrt{72} \cdot \sqrt{x^2}\)[/tex]
- [tex]\(\sqrt{72}\)[/tex] can be simplified as [tex]\(\sqrt{36 \cdot 2} = \sqrt{36} \cdot \sqrt{2} = 6\sqrt{2}\)[/tex]
- [tex]\(\sqrt{x^2} = x\)[/tex]
- Thus, [tex]\(\sqrt{72 x^2} = 6\sqrt{2} \cdot x\)[/tex]
Now, let's compare the simplified forms to check for like radicals:
1. [tex]\(\sqrt{50 x^2}\)[/tex] simplifies to [tex]\(5\sqrt{2} \cdot x\)[/tex]
2. [tex]\(\sqrt{32 x}\)[/tex] simplifies to [tex]\(4\sqrt{2} \cdot \sqrt{x}\)[/tex]
3. [tex]\(\sqrt{18 n}\)[/tex] simplifies to [tex]\(3\sqrt{2} \cdot \sqrt{n}\)[/tex]
4. [tex]\(\sqrt{72 x^2}\)[/tex] simplifies to [tex]\(6\sqrt{2} \cdot x\)[/tex]
For radicals to be like, their radicands must be identical. In this case:
- [tex]\(5\sqrt{2} \cdot x\)[/tex] and [tex]\(6\sqrt{2} \cdot x\)[/tex] both have [tex]\(\sqrt{2}\cdot x\)[/tex] as a part of the expression but differ by coefficients 5 and 6. Despite simplifying to show similar forms internally, they cannot be directly combined as 'like' radicals.
Hence, none of these simplified radical expressions are like radicals after simplification.
