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Sagot :
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.
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.
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