Answer :
To solve the problem of finding which choice is equivalent to the expression [tex]\(\sqrt{45} + \sqrt{125}\)[/tex], we will simplify the square roots:
1. Simplify [tex]\(\sqrt{45}\)[/tex]:
- Recognize that 45 can be factored into [tex]\(9 \times 5\)[/tex].
- Thus, [tex]\(\sqrt{45} = \sqrt{9 \times 5} = \sqrt{9} \times \sqrt{5}\)[/tex].
- Calculate [tex]\(\sqrt{9} = 3\)[/tex].
- This gives us [tex]\(\sqrt{45} = 3\sqrt{5}\)[/tex].
2. Simplify [tex]\(\sqrt{125}\)[/tex]:
- Recognize that 125 can be factored into [tex]\(25 \times 5\)[/tex].
- Thus, [tex]\(\sqrt{125} = \sqrt{25 \times 5} = \sqrt{25} \times \sqrt{5}\)[/tex].
- Calculate [tex]\(\sqrt{25} = 5\)[/tex].
- This gives us [tex]\(\sqrt{125} = 5\sqrt{5}\)[/tex].
3. Combine the results:
- Add the simplified expressions: [tex]\(3\sqrt{5} + 5\sqrt{5}\)[/tex].
- Combine like terms: [tex]\( (3 + 5)\sqrt{5} = 8\sqrt{5}\)[/tex].
The equivalence of the expression [tex]\(\sqrt{45} + \sqrt{125}\)[/tex] is [tex]\(8\sqrt{5}\)[/tex]. Therefore, the correct choice is:
C. [tex]\(8\sqrt{5}\)[/tex]
1. Simplify [tex]\(\sqrt{45}\)[/tex]:
- Recognize that 45 can be factored into [tex]\(9 \times 5\)[/tex].
- Thus, [tex]\(\sqrt{45} = \sqrt{9 \times 5} = \sqrt{9} \times \sqrt{5}\)[/tex].
- Calculate [tex]\(\sqrt{9} = 3\)[/tex].
- This gives us [tex]\(\sqrt{45} = 3\sqrt{5}\)[/tex].
2. Simplify [tex]\(\sqrt{125}\)[/tex]:
- Recognize that 125 can be factored into [tex]\(25 \times 5\)[/tex].
- Thus, [tex]\(\sqrt{125} = \sqrt{25 \times 5} = \sqrt{25} \times \sqrt{5}\)[/tex].
- Calculate [tex]\(\sqrt{25} = 5\)[/tex].
- This gives us [tex]\(\sqrt{125} = 5\sqrt{5}\)[/tex].
3. Combine the results:
- Add the simplified expressions: [tex]\(3\sqrt{5} + 5\sqrt{5}\)[/tex].
- Combine like terms: [tex]\( (3 + 5)\sqrt{5} = 8\sqrt{5}\)[/tex].
The equivalence of the expression [tex]\(\sqrt{45} + \sqrt{125}\)[/tex] is [tex]\(8\sqrt{5}\)[/tex]. Therefore, the correct choice is:
C. [tex]\(8\sqrt{5}\)[/tex]