Select all that are like radicals after simplifying:

[tex]\[ \sqrt{50 x^2} \][/tex]

[tex]\[ \sqrt{32 x} \][/tex]

[tex]\[ \sqrt{18 n} \][/tex]

[tex]\[ \sqrt{72 x^2} \][/tex]



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].