Answer :
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{25 \cdot 2 \cdot x^2} = \sqrt{25} \cdot \sqrt{2} \cdot \sqrt{x^2} = 5 \cdot \sqrt{2} \cdot |x| = 5 \cdot \sqrt{2} \cdot x \quad \text{(assuming \(x\) is non-negative)} \][/tex]
Thus, [tex]\(\sqrt{50 x^2}\)[/tex] simplifies to [tex]\(5\sqrt{2} \cdot x\)[/tex].
2. Simplify [tex]\(\sqrt{32 x}\)[/tex]:
[tex]\[ \sqrt{32 x} = \sqrt{16 \cdot 2 \cdot x} = \sqrt{16} \cdot \sqrt{2} \cdot \sqrt{x} = 4 \cdot \sqrt{2} \cdot \sqrt{x} \][/tex]
Thus, [tex]\(\sqrt{32 x}\)[/tex] simplifies to [tex]\(4\sqrt{2} \cdot \sqrt{x}\)[/tex].
3. Simplify [tex]\(\sqrt{18 n}\)[/tex]:
[tex]\[ \sqrt{18 n} = \sqrt{9 \cdot 2 \cdot n} = \sqrt{9} \cdot \sqrt{2} \cdot \sqrt{n} = 3 \cdot \sqrt{2} \cdot \sqrt{n} \][/tex]
Thus, [tex]\(\sqrt{18 n}\)[/tex] simplifies to [tex]\(3\sqrt{2} \cdot \sqrt{n}\)[/tex].
4. Simplify [tex]\(\sqrt{72 x^2}\)[/tex]:
[tex]\[ \sqrt{72 x^2} = \sqrt{36 \cdot 2 \cdot x^2} = \sqrt{36} \cdot \sqrt{2} \cdot \sqrt{x^2} = 6 \cdot \sqrt{2} \cdot |x| = 6 \cdot \sqrt{2} \cdot x \quad \text{(assuming \(x\) is non-negative)} \][/tex]
Thus, [tex]\(\sqrt{72 x^2}\)[/tex] simplifies to [tex]\(6\sqrt{2} \cdot x\)[/tex].
Now let's compare the simplified forms:
1. [tex]\(5\sqrt{2} \cdot x\)[/tex]
2. [tex]\(4\sqrt{2} \cdot \sqrt{x}\)[/tex]
3. [tex]\(3\sqrt{2} \cdot \sqrt{n}\)[/tex]
4. [tex]\(6\sqrt{2} \cdot x\)[/tex]
Like radicals are those that have the same radicand and the same exponent. Here, [tex]\(5\sqrt{2} \cdot x\)[/tex] and [tex]\(6\sqrt{2} \cdot x\)[/tex] are like radicals because they both involve [tex]\(\sqrt{2} \cdot x\)[/tex]. The other two expressions, [tex]\(4\sqrt{2} \cdot \sqrt{x}\)[/tex] and [tex]\(3\sqrt{2} \cdot \sqrt{n}\)[/tex], do not match.
Thus, the expressions containing like radicals after simplifying are:
[tex]\[ 5\sqrt{2} \cdot x \quad \text{and} \quad 6\sqrt{2} \cdot x. \][/tex]
The corresponding original expressions are:
[tex]\(\sqrt{50 x^2}\)[/tex] and [tex]\(\sqrt{72 x^2}\)[/tex].
1. Simplify [tex]\(\sqrt{50 x^2}\)[/tex]:
[tex]\[ \sqrt{50 x^2} = \sqrt{25 \cdot 2 \cdot x^2} = \sqrt{25} \cdot \sqrt{2} \cdot \sqrt{x^2} = 5 \cdot \sqrt{2} \cdot |x| = 5 \cdot \sqrt{2} \cdot x \quad \text{(assuming \(x\) is non-negative)} \][/tex]
Thus, [tex]\(\sqrt{50 x^2}\)[/tex] simplifies to [tex]\(5\sqrt{2} \cdot x\)[/tex].
2. Simplify [tex]\(\sqrt{32 x}\)[/tex]:
[tex]\[ \sqrt{32 x} = \sqrt{16 \cdot 2 \cdot x} = \sqrt{16} \cdot \sqrt{2} \cdot \sqrt{x} = 4 \cdot \sqrt{2} \cdot \sqrt{x} \][/tex]
Thus, [tex]\(\sqrt{32 x}\)[/tex] simplifies to [tex]\(4\sqrt{2} \cdot \sqrt{x}\)[/tex].
3. Simplify [tex]\(\sqrt{18 n}\)[/tex]:
[tex]\[ \sqrt{18 n} = \sqrt{9 \cdot 2 \cdot n} = \sqrt{9} \cdot \sqrt{2} \cdot \sqrt{n} = 3 \cdot \sqrt{2} \cdot \sqrt{n} \][/tex]
Thus, [tex]\(\sqrt{18 n}\)[/tex] simplifies to [tex]\(3\sqrt{2} \cdot \sqrt{n}\)[/tex].
4. Simplify [tex]\(\sqrt{72 x^2}\)[/tex]:
[tex]\[ \sqrt{72 x^2} = \sqrt{36 \cdot 2 \cdot x^2} = \sqrt{36} \cdot \sqrt{2} \cdot \sqrt{x^2} = 6 \cdot \sqrt{2} \cdot |x| = 6 \cdot \sqrt{2} \cdot x \quad \text{(assuming \(x\) is non-negative)} \][/tex]
Thus, [tex]\(\sqrt{72 x^2}\)[/tex] simplifies to [tex]\(6\sqrt{2} \cdot x\)[/tex].
Now let's compare the simplified forms:
1. [tex]\(5\sqrt{2} \cdot x\)[/tex]
2. [tex]\(4\sqrt{2} \cdot \sqrt{x}\)[/tex]
3. [tex]\(3\sqrt{2} \cdot \sqrt{n}\)[/tex]
4. [tex]\(6\sqrt{2} \cdot x\)[/tex]
Like radicals are those that have the same radicand and the same exponent. Here, [tex]\(5\sqrt{2} \cdot x\)[/tex] and [tex]\(6\sqrt{2} \cdot x\)[/tex] are like radicals because they both involve [tex]\(\sqrt{2} \cdot x\)[/tex]. The other two expressions, [tex]\(4\sqrt{2} \cdot \sqrt{x}\)[/tex] and [tex]\(3\sqrt{2} \cdot \sqrt{n}\)[/tex], do not match.
Thus, the expressions containing like radicals after simplifying are:
[tex]\[ 5\sqrt{2} \cdot x \quad \text{and} \quad 6\sqrt{2} \cdot x. \][/tex]
The corresponding original expressions are:
[tex]\(\sqrt{50 x^2}\)[/tex] and [tex]\(\sqrt{72 x^2}\)[/tex].
