To determine which of the given radicals are like radicals after simplifying, let's simplify each expression step-by-step.
1. [tex]\(\sqrt{50 x^2}\)[/tex]:
[tex]\[
\sqrt{50 x^2} = \sqrt{50} \cdot \sqrt{x^2}
\][/tex]
Since [tex]\(\sqrt{x^2} = x\)[/tex], we get:
[tex]\[
\sqrt{50} \cdot x
\][/tex]
Next, simplify [tex]\(\sqrt{50}\)[/tex]:
[tex]\[
\sqrt{50} = \sqrt{25 \cdot 2} = \sqrt{25} \cdot \sqrt{2} = 5\sqrt{2}
\][/tex]
Therefore,
[tex]\[
\sqrt{50 x^2} = 5x\sqrt{2}
\][/tex]
2. [tex]\(\sqrt{32 x}\)[/tex]:
[tex]\[
\sqrt{32 x} = \sqrt{32} \cdot \sqrt{x}
\][/tex]
Simplify [tex]\(\sqrt{32}\)[/tex]:
[tex]\[
\sqrt{32} = \sqrt{16 \cdot 2} = \sqrt{16} \cdot \sqrt{2} = 4\sqrt{2}
\][/tex]
Therefore,
[tex]\[
\sqrt{32 x} = 4\sqrt{2} \cdot \sqrt{x} = 4\sqrt{2x}
\][/tex]
3. [tex]\(\sqrt{18 n}\)[/tex]:
[tex]\[
\sqrt{18 n} = \sqrt{18} \cdot \sqrt{n}
\][/tex]
Simplify [tex]\(\sqrt{18}\)[/tex]:
[tex]\[
\sqrt{18} = \sqrt{9 \cdot 2} = \sqrt{9} \cdot \sqrt{2} = 3\sqrt{2}
\][/tex]
Therefore,
[tex]\[
\sqrt{18 n} = 3\sqrt{2} \cdot \sqrt{n} = 3\sqrt{2n}
\][/tex]
4. [tex]\(\sqrt{72 x^2}\)[/tex]:
[tex]\[
\sqrt{72 x^2} = \sqrt{72} \cdot \sqrt{x^2}
\][/tex]
Since [tex]\(\sqrt{x^2} = x\)[/tex], we get:
[tex]\[
\sqrt{72} \cdot x
\][/tex]
Simplify [tex]\(\sqrt{72}\)[/tex]:
[tex]\[
\sqrt{72} = \sqrt{36 \cdot 2} = \sqrt{36} \cdot \sqrt{2} = 6\sqrt{2}
\][/tex]
Therefore,
[tex]\[
\sqrt{72 x^2} = 6x\sqrt{2}
\][/tex]
After simplifying, we have:
- [tex]\(\sqrt{50 x^2} = 5x\sqrt{2}\)[/tex]
- [tex]\(\sqrt{32 x} = 4\sqrt{2x}\)[/tex]
- [tex]\(\sqrt{18 n} = 3\sqrt{2n}\)[/tex]
- [tex]\(\sqrt{72 x^2} = 6x\sqrt{2}\)[/tex]
Among these, the simplified forms [tex]\(5x\sqrt{2}\)[/tex] and [tex]\(6x\sqrt{2}\)[/tex] have the [tex]\(\sqrt{2}\)[/tex] term, and the variable [tex]\(x\)[/tex] outside the radical. Hence, the like radicals are:
[tex]\(\sqrt{50 x^2}\)[/tex] and [tex]\(\sqrt{72 x^2}\)[/tex].
