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Sagot :
To determine which expressions are like radicals after simplifying, we first need to simplify each of the given radicals individually.
1. Simplifying [tex]\(\sqrt{50 x^2}\)[/tex]:
Observe that [tex]\(50 = 25 \cdot 2\)[/tex]. Therefore:
[tex]\[ \sqrt{50 x^2} = \sqrt{25 \cdot 2 \cdot x^2} \][/tex]
Using the property [tex]\(\sqrt{a \cdot b} = \sqrt{a} \cdot \sqrt{b}\)[/tex]:
[tex]\[ \sqrt{25 \cdot 2 \cdot x^2} = \sqrt{25} \cdot \sqrt{2} \cdot \sqrt{x^2} \][/tex]
Simplify each term:
[tex]\[ \sqrt{25} = 5, \quad \sqrt{2} \text{ remains as } \sqrt{2}, \quad \sqrt{x^2} = |x| \][/tex]
Therefore:
[tex]\[ \sqrt{50 x^2} = 5 \sqrt{2} \cdot |x| \][/tex]
2. Simplifying [tex]\(\sqrt{32 x}\)[/tex]:
Observe that [tex]\(32 = 16 \cdot 2\)[/tex]. Therefore:
[tex]\[ \sqrt{32 x} = \sqrt{16 \cdot 2 \cdot x} \][/tex]
Using the property [tex]\(\sqrt{a \cdot b} = \sqrt{a} \cdot \sqrt{b}\)[/tex]:
[tex]\[ \sqrt{16 \cdot 2 \cdot x} = \sqrt{16} \cdot \sqrt{2} \cdot \sqrt{x} \][/tex]
Simplify each term:
[tex]\[ \sqrt{16} = 4, \quad \sqrt{2} \text{ remains as } \sqrt{2}, \quad \sqrt{x} = \sqrt{x} \][/tex]
Therefore:
[tex]\[ \sqrt{32 x} = 4 \sqrt{2} \cdot \sqrt{x} \][/tex]
3. Simplifying [tex]\(\sqrt{18 n}\)[/tex]:
Observe that [tex]\(18 = 9 \cdot 2\)[/tex]. Therefore:
[tex]\[ \sqrt{18 n} = \sqrt{9 \cdot 2 \cdot n} \][/tex]
Using the property [tex]\(\sqrt{a \cdot b} = \sqrt{a} \cdot \sqrt{b}\)[/tex]:
[tex]\[ \sqrt{9 \cdot 2 \cdot n} = \sqrt{9} \cdot \sqrt{2} \cdot \sqrt{n} \][/tex]
Simplify each term:
[tex]\[ \sqrt{9} = 3, \quad \sqrt{2} \text{ remains as } \sqrt{2}, \quad \sqrt{n} = \sqrt{n} \][/tex]
Therefore:
[tex]\[ \sqrt{18 n} = 3 \sqrt{2} \cdot \sqrt{n} \][/tex]
4. Simplifying [tex]\(\sqrt{72 x^2}\)[/tex]:
Observe that [tex]\(72 = 36 \cdot 2\)[/tex]. Therefore:
[tex]\[ \sqrt{72 x^2} = \sqrt{36 \cdot 2 \cdot x^2} \][/tex]
Using the property [tex]\(\sqrt{a \cdot b} = \sqrt{a} \cdot \sqrt{b}\)[/tex]:
[tex]\[ \sqrt{36 \cdot 2 \cdot x^2} = \sqrt{36} \cdot \sqrt{2} \cdot \sqrt{x^2} \][/tex]
Simplify each term:
[tex]\[ \sqrt{36} = 6, \quad \sqrt{2} \text{ remains as } \sqrt{2}, \quad \sqrt{x^2} = x \][/tex]
Therefore:
[tex]\[ \sqrt{72 x^2} = 6 \sqrt{2} \cdot x \][/tex]
Now, let's compile our simplified expressions:
1. [tex]\(\sqrt{50 x^2} = 5 \sqrt{2} \cdot |x|\)[/tex]
2. [tex]\(\sqrt{32 x} = 4 \sqrt{2} \cdot \sqrt{x}\)[/tex]
3. [tex]\(\sqrt{18 n} = 3 \sqrt{2} \cdot \sqrt{n}\)[/tex]
4. [tex]\(\sqrt{72 x^2} = 6 \sqrt{2} \cdot x\)[/tex]
To find like radicals, we need them to have the same radicand. The simplified forms have [tex]\(\sqrt{2}\)[/tex] as a common factor. However, we also need to consider the entire expression.
After comparing:
- [tex]\(\sqrt{50 x^2} = 5 \sqrt{2} \cdot |x|\)[/tex]
- [tex]\(\sqrt{72 x^2} = 6 \sqrt{2} \cdot x\)[/tex]
Both of these expressions involve [tex]\(x^2\)[/tex] under the radical, making them like radicals. Meanwhile, the others do not have the same structure.
Thus, the like radicals after simplifying are:
[tex]\[ \boxed{\sqrt{50 x^2} \text{ and } \sqrt{72 x^2}} \][/tex]
1. Simplifying [tex]\(\sqrt{50 x^2}\)[/tex]:
Observe that [tex]\(50 = 25 \cdot 2\)[/tex]. Therefore:
[tex]\[ \sqrt{50 x^2} = \sqrt{25 \cdot 2 \cdot x^2} \][/tex]
Using the property [tex]\(\sqrt{a \cdot b} = \sqrt{a} \cdot \sqrt{b}\)[/tex]:
[tex]\[ \sqrt{25 \cdot 2 \cdot x^2} = \sqrt{25} \cdot \sqrt{2} \cdot \sqrt{x^2} \][/tex]
Simplify each term:
[tex]\[ \sqrt{25} = 5, \quad \sqrt{2} \text{ remains as } \sqrt{2}, \quad \sqrt{x^2} = |x| \][/tex]
Therefore:
[tex]\[ \sqrt{50 x^2} = 5 \sqrt{2} \cdot |x| \][/tex]
2. Simplifying [tex]\(\sqrt{32 x}\)[/tex]:
Observe that [tex]\(32 = 16 \cdot 2\)[/tex]. Therefore:
[tex]\[ \sqrt{32 x} = \sqrt{16 \cdot 2 \cdot x} \][/tex]
Using the property [tex]\(\sqrt{a \cdot b} = \sqrt{a} \cdot \sqrt{b}\)[/tex]:
[tex]\[ \sqrt{16 \cdot 2 \cdot x} = \sqrt{16} \cdot \sqrt{2} \cdot \sqrt{x} \][/tex]
Simplify each term:
[tex]\[ \sqrt{16} = 4, \quad \sqrt{2} \text{ remains as } \sqrt{2}, \quad \sqrt{x} = \sqrt{x} \][/tex]
Therefore:
[tex]\[ \sqrt{32 x} = 4 \sqrt{2} \cdot \sqrt{x} \][/tex]
3. Simplifying [tex]\(\sqrt{18 n}\)[/tex]:
Observe that [tex]\(18 = 9 \cdot 2\)[/tex]. Therefore:
[tex]\[ \sqrt{18 n} = \sqrt{9 \cdot 2 \cdot n} \][/tex]
Using the property [tex]\(\sqrt{a \cdot b} = \sqrt{a} \cdot \sqrt{b}\)[/tex]:
[tex]\[ \sqrt{9 \cdot 2 \cdot n} = \sqrt{9} \cdot \sqrt{2} \cdot \sqrt{n} \][/tex]
Simplify each term:
[tex]\[ \sqrt{9} = 3, \quad \sqrt{2} \text{ remains as } \sqrt{2}, \quad \sqrt{n} = \sqrt{n} \][/tex]
Therefore:
[tex]\[ \sqrt{18 n} = 3 \sqrt{2} \cdot \sqrt{n} \][/tex]
4. Simplifying [tex]\(\sqrt{72 x^2}\)[/tex]:
Observe that [tex]\(72 = 36 \cdot 2\)[/tex]. Therefore:
[tex]\[ \sqrt{72 x^2} = \sqrt{36 \cdot 2 \cdot x^2} \][/tex]
Using the property [tex]\(\sqrt{a \cdot b} = \sqrt{a} \cdot \sqrt{b}\)[/tex]:
[tex]\[ \sqrt{36 \cdot 2 \cdot x^2} = \sqrt{36} \cdot \sqrt{2} \cdot \sqrt{x^2} \][/tex]
Simplify each term:
[tex]\[ \sqrt{36} = 6, \quad \sqrt{2} \text{ remains as } \sqrt{2}, \quad \sqrt{x^2} = x \][/tex]
Therefore:
[tex]\[ \sqrt{72 x^2} = 6 \sqrt{2} \cdot x \][/tex]
Now, let's compile our simplified expressions:
1. [tex]\(\sqrt{50 x^2} = 5 \sqrt{2} \cdot |x|\)[/tex]
2. [tex]\(\sqrt{32 x} = 4 \sqrt{2} \cdot \sqrt{x}\)[/tex]
3. [tex]\(\sqrt{18 n} = 3 \sqrt{2} \cdot \sqrt{n}\)[/tex]
4. [tex]\(\sqrt{72 x^2} = 6 \sqrt{2} \cdot x\)[/tex]
To find like radicals, we need them to have the same radicand. The simplified forms have [tex]\(\sqrt{2}\)[/tex] as a common factor. However, we also need to consider the entire expression.
After comparing:
- [tex]\(\sqrt{50 x^2} = 5 \sqrt{2} \cdot |x|\)[/tex]
- [tex]\(\sqrt{72 x^2} = 6 \sqrt{2} \cdot x\)[/tex]
Both of these expressions involve [tex]\(x^2\)[/tex] under the radical, making them like radicals. Meanwhile, the others do not have the same structure.
Thus, the like radicals after simplifying are:
[tex]\[ \boxed{\sqrt{50 x^2} \text{ and } \sqrt{72 x^2}} \][/tex]
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