Answer :
Let's match each of the given radicals to their simplified forms step-by-step.
### Step-by-Step Solution:
#### Matching a: [tex]\(\sqrt{-50}\)[/tex]
1. To simplify [tex]\(\sqrt{-50}\)[/tex]:
- Introduce the imaginary unit: [tex]\( \sqrt{-50} = \sqrt{50} \cdot \sqrt{-1} = \sqrt{50} \cdot i \)[/tex].
- Simplify [tex]\(\sqrt{50}\)[/tex]: [tex]\( \sqrt{50} = \sqrt{25 \cdot 2} = \sqrt{25} \cdot \sqrt{2} = 5 \sqrt{2} \)[/tex].
- Therefore, [tex]\(\sqrt{-50} = 5 i \sqrt{2}\)[/tex].
#### Matching b: [tex]\(\sqrt{-252}\)[/tex]
2. To simplify [tex]\(\sqrt{-252}\)[/tex]:
- Introduce the imaginary unit: [tex]\( \sqrt{-252} = \sqrt{252} \cdot \sqrt{-1} = \sqrt{252} \cdot i \)[/tex].
- Simplify [tex]\(\sqrt{252}\)[/tex]: [tex]\( \sqrt{252} = \sqrt{36 \cdot 7} = \sqrt{36} \cdot \sqrt{7} = 6 \sqrt{7} \)[/tex].
- Therefore, [tex]\(\sqrt{-252} = 6 i \sqrt{7}\)[/tex].
#### Matching c: [tex]\(4 \sqrt{54}\)[/tex]
3. To simplify [tex]\(4 \sqrt{54}\)[/tex]:
- Simplify [tex]\(\sqrt{54}\)[/tex]: [tex]\( \sqrt{54} = \sqrt{9 \cdot 6} = \sqrt{9} \cdot \sqrt{6} = 3 \sqrt{6} \)[/tex].
- Therefore, [tex]\(4 \sqrt{54} = 4 \cdot 3 \sqrt{6} = 12 \sqrt{6}\)[/tex].
#### Matching d: [tex]\(\sqrt{-81}\)[/tex]
4. To simplify [tex]\(\sqrt{-81}\)[/tex]:
- Introduce the imaginary unit: [tex]\( \sqrt{-81} = \sqrt{81} \cdot \sqrt{-1} = \sqrt{81} \cdot i \)[/tex].
- Simplify [tex]\(\sqrt{81}\)[/tex]: [tex]\( \sqrt{81} = 9 \)[/tex].
- Therefore, [tex]\(\sqrt{-81} = 9 i\)[/tex].
### Final Matches:
a. [tex]\(\sqrt{-50}\)[/tex] matches with [tex]\(5 i \sqrt{2}\)[/tex] (option 1).
b. [tex]\(\sqrt{-252}\)[/tex] matches with [tex]\(6 i \sqrt{7}\)[/tex] (option 4).
c. [tex]\(4 \sqrt{54}\)[/tex] matches with [tex]\(12 \sqrt{6}\)[/tex] (option 2).
d. [tex]\(\sqrt{-81}\)[/tex] matches with [tex]\(9 i\)[/tex] (option 3).
Thus, the correct matches are summarized as follows:
- a. [tex]\(\sqrt{-50}\)[/tex] = 5 i [tex]\(\sqrt{2}\)[/tex] (option 1)
- b. [tex]\(\sqrt{-252}\)[/tex] = 6 i [tex]\(\sqrt{7}\)[/tex] (option 4)
- c. [tex]\(4 \sqrt{54}\)[/tex] = 12 [tex]\(\sqrt{6}\)[/tex] (option 2)
- d. [tex]\(\sqrt{-81}\)[/tex] = 9 i (option 3)
The matches are: [tex]\( (1, 4, 2, 3) \)[/tex].
### Step-by-Step Solution:
#### Matching a: [tex]\(\sqrt{-50}\)[/tex]
1. To simplify [tex]\(\sqrt{-50}\)[/tex]:
- Introduce the imaginary unit: [tex]\( \sqrt{-50} = \sqrt{50} \cdot \sqrt{-1} = \sqrt{50} \cdot i \)[/tex].
- Simplify [tex]\(\sqrt{50}\)[/tex]: [tex]\( \sqrt{50} = \sqrt{25 \cdot 2} = \sqrt{25} \cdot \sqrt{2} = 5 \sqrt{2} \)[/tex].
- Therefore, [tex]\(\sqrt{-50} = 5 i \sqrt{2}\)[/tex].
#### Matching b: [tex]\(\sqrt{-252}\)[/tex]
2. To simplify [tex]\(\sqrt{-252}\)[/tex]:
- Introduce the imaginary unit: [tex]\( \sqrt{-252} = \sqrt{252} \cdot \sqrt{-1} = \sqrt{252} \cdot i \)[/tex].
- Simplify [tex]\(\sqrt{252}\)[/tex]: [tex]\( \sqrt{252} = \sqrt{36 \cdot 7} = \sqrt{36} \cdot \sqrt{7} = 6 \sqrt{7} \)[/tex].
- Therefore, [tex]\(\sqrt{-252} = 6 i \sqrt{7}\)[/tex].
#### Matching c: [tex]\(4 \sqrt{54}\)[/tex]
3. To simplify [tex]\(4 \sqrt{54}\)[/tex]:
- Simplify [tex]\(\sqrt{54}\)[/tex]: [tex]\( \sqrt{54} = \sqrt{9 \cdot 6} = \sqrt{9} \cdot \sqrt{6} = 3 \sqrt{6} \)[/tex].
- Therefore, [tex]\(4 \sqrt{54} = 4 \cdot 3 \sqrt{6} = 12 \sqrt{6}\)[/tex].
#### Matching d: [tex]\(\sqrt{-81}\)[/tex]
4. To simplify [tex]\(\sqrt{-81}\)[/tex]:
- Introduce the imaginary unit: [tex]\( \sqrt{-81} = \sqrt{81} \cdot \sqrt{-1} = \sqrt{81} \cdot i \)[/tex].
- Simplify [tex]\(\sqrt{81}\)[/tex]: [tex]\( \sqrt{81} = 9 \)[/tex].
- Therefore, [tex]\(\sqrt{-81} = 9 i\)[/tex].
### Final Matches:
a. [tex]\(\sqrt{-50}\)[/tex] matches with [tex]\(5 i \sqrt{2}\)[/tex] (option 1).
b. [tex]\(\sqrt{-252}\)[/tex] matches with [tex]\(6 i \sqrt{7}\)[/tex] (option 4).
c. [tex]\(4 \sqrt{54}\)[/tex] matches with [tex]\(12 \sqrt{6}\)[/tex] (option 2).
d. [tex]\(\sqrt{-81}\)[/tex] matches with [tex]\(9 i\)[/tex] (option 3).
Thus, the correct matches are summarized as follows:
- a. [tex]\(\sqrt{-50}\)[/tex] = 5 i [tex]\(\sqrt{2}\)[/tex] (option 1)
- b. [tex]\(\sqrt{-252}\)[/tex] = 6 i [tex]\(\sqrt{7}\)[/tex] (option 4)
- c. [tex]\(4 \sqrt{54}\)[/tex] = 12 [tex]\(\sqrt{6}\)[/tex] (option 2)
- d. [tex]\(\sqrt{-81}\)[/tex] = 9 i (option 3)
The matches are: [tex]\( (1, 4, 2, 3) \)[/tex].
