Answer :
Let's simplify the expression [tex]\(\sqrt{40} + 2 \sqrt{10} + \sqrt{90}\)[/tex] step by step.
1. Simplify [tex]\(\sqrt{40}\)[/tex]:
- Break down 40 into its factors: [tex]\(40 = 4 \times 10\)[/tex].
- We can take the square root of 4, which is 2, out of the square root: [tex]\(\sqrt{40} = \sqrt{4 \times 10} = 2\sqrt{10}\)[/tex].
2. Simplify [tex]\(2\sqrt{10}\)[/tex]:
- This term is already in its simplest form, so it remains [tex]\(2\sqrt{10}\)[/tex].
3. Simplify [tex]\(\sqrt{90}\)[/tex]:
- Break down 90 into its factors: [tex]\(90 = 9 \times 10\)[/tex].
- We can take the square root of 9, which is 3, out of the square root: [tex]\(\sqrt{90} = \sqrt{9 \times 10} = 3\sqrt{10}\)[/tex].
4. Combine all terms:
- Now we have: [tex]\(2\sqrt{10} + 2\sqrt{10} + 3\sqrt{10}\)[/tex].
- Add the coefficients of [tex]\(\sqrt{10}\)[/tex]: [tex]\(2 + 2 + 3 = 7\)[/tex].
- So, the expression simplifies to [tex]\(7\sqrt{10}\)[/tex].
Therefore, the expression [tex]\(\sqrt{40} + 2 \sqrt{10} + \sqrt{90}\)[/tex] is equivalent to [tex]\(\boxed{7\sqrt{10}}\)[/tex], which corresponds to option A.
1. Simplify [tex]\(\sqrt{40}\)[/tex]:
- Break down 40 into its factors: [tex]\(40 = 4 \times 10\)[/tex].
- We can take the square root of 4, which is 2, out of the square root: [tex]\(\sqrt{40} = \sqrt{4 \times 10} = 2\sqrt{10}\)[/tex].
2. Simplify [tex]\(2\sqrt{10}\)[/tex]:
- This term is already in its simplest form, so it remains [tex]\(2\sqrt{10}\)[/tex].
3. Simplify [tex]\(\sqrt{90}\)[/tex]:
- Break down 90 into its factors: [tex]\(90 = 9 \times 10\)[/tex].
- We can take the square root of 9, which is 3, out of the square root: [tex]\(\sqrt{90} = \sqrt{9 \times 10} = 3\sqrt{10}\)[/tex].
4. Combine all terms:
- Now we have: [tex]\(2\sqrt{10} + 2\sqrt{10} + 3\sqrt{10}\)[/tex].
- Add the coefficients of [tex]\(\sqrt{10}\)[/tex]: [tex]\(2 + 2 + 3 = 7\)[/tex].
- So, the expression simplifies to [tex]\(7\sqrt{10}\)[/tex].
Therefore, the expression [tex]\(\sqrt{40} + 2 \sqrt{10} + \sqrt{90}\)[/tex] is equivalent to [tex]\(\boxed{7\sqrt{10}}\)[/tex], which corresponds to option A.