Answer :
To solve the expression [tex]\(\sqrt{40} + 8\sqrt{10} + \sqrt{90}\)[/tex], we need to simplify each square root term.
1. Simplify [tex]\(\sqrt{40}\)[/tex]:
- Break down 40 into its factors: [tex]\(40 = 4 \times 10\)[/tex].
- Since 4 is a perfect square, we can simplify: [tex]\(\sqrt{40} = \sqrt{4 \times 10} = \sqrt{4} \times \sqrt{10} = 2\sqrt{10}\)[/tex].
2. Leave [tex]\(8\sqrt{10}\)[/tex] as it is:
- This term is already in its simplest form.
3. Simplify [tex]\(\sqrt{90}\)[/tex]:
- Break down 90 into its factors: [tex]\(90 = 9 \times 10\)[/tex].
- Since 9 is a perfect square, we can simplify: [tex]\(\sqrt{90} = \sqrt{9 \times 10} = \sqrt{9} \times \sqrt{10} = 3\sqrt{10}\)[/tex].
4. Combine all the terms:
- Now we have the simplified terms: [tex]\(2\sqrt{10} + 8\sqrt{10} + 3\sqrt{10}\)[/tex].
- Combine the coefficients of [tex]\(\sqrt{10}\)[/tex]: [tex]\(2 + 8 + 3 = 13\)[/tex].
Therefore, the equivalent expression is [tex]\(13\sqrt{10}\)[/tex].
The correct choice is D. [tex]\(13\sqrt{10}\)[/tex].
1. Simplify [tex]\(\sqrt{40}\)[/tex]:
- Break down 40 into its factors: [tex]\(40 = 4 \times 10\)[/tex].
- Since 4 is a perfect square, we can simplify: [tex]\(\sqrt{40} = \sqrt{4 \times 10} = \sqrt{4} \times \sqrt{10} = 2\sqrt{10}\)[/tex].
2. Leave [tex]\(8\sqrt{10}\)[/tex] as it is:
- This term is already in its simplest form.
3. Simplify [tex]\(\sqrt{90}\)[/tex]:
- Break down 90 into its factors: [tex]\(90 = 9 \times 10\)[/tex].
- Since 9 is a perfect square, we can simplify: [tex]\(\sqrt{90} = \sqrt{9 \times 10} = \sqrt{9} \times \sqrt{10} = 3\sqrt{10}\)[/tex].
4. Combine all the terms:
- Now we have the simplified terms: [tex]\(2\sqrt{10} + 8\sqrt{10} + 3\sqrt{10}\)[/tex].
- Combine the coefficients of [tex]\(\sqrt{10}\)[/tex]: [tex]\(2 + 8 + 3 = 13\)[/tex].
Therefore, the equivalent expression is [tex]\(13\sqrt{10}\)[/tex].
The correct choice is D. [tex]\(13\sqrt{10}\)[/tex].