Answer :
Let's simplify the expression [tex]\(\sqrt{40} + 8\sqrt{10} + \sqrt{90}\)[/tex].
1. Simplify [tex]\(\sqrt{40}\)[/tex]:
[tex]\(\sqrt{40} = \sqrt{4 \times 10} = \sqrt{4} \times \sqrt{10} = 2\sqrt{10}\)[/tex].
2. Simplify [tex]\(\sqrt{90}\)[/tex]:
[tex]\(\sqrt{90} = \sqrt{9 \times 10} = \sqrt{9} \times \sqrt{10} = 3\sqrt{10}\)[/tex].
3. Combine all the terms:
Now substituting back the simplified terms:
[tex]\[
\sqrt{40} + 8\sqrt{10} + \sqrt{90} = 2\sqrt{10} + 8\sqrt{10} + 3\sqrt{10}
\][/tex]
All terms are like terms because they each contain [tex]\(\sqrt{10}\)[/tex]. Add the coefficients together:
[tex]\[
(2 + 8 + 3)\sqrt{10} = 13\sqrt{10}
\][/tex]
Therefore, the expression simplifies to [tex]\(13\sqrt{10}\)[/tex].
The correct choice is C: [tex]\(13\sqrt{10}\)[/tex].
1. Simplify [tex]\(\sqrt{40}\)[/tex]:
[tex]\(\sqrt{40} = \sqrt{4 \times 10} = \sqrt{4} \times \sqrt{10} = 2\sqrt{10}\)[/tex].
2. Simplify [tex]\(\sqrt{90}\)[/tex]:
[tex]\(\sqrt{90} = \sqrt{9 \times 10} = \sqrt{9} \times \sqrt{10} = 3\sqrt{10}\)[/tex].
3. Combine all the terms:
Now substituting back the simplified terms:
[tex]\[
\sqrt{40} + 8\sqrt{10} + \sqrt{90} = 2\sqrt{10} + 8\sqrt{10} + 3\sqrt{10}
\][/tex]
All terms are like terms because they each contain [tex]\(\sqrt{10}\)[/tex]. Add the coefficients together:
[tex]\[
(2 + 8 + 3)\sqrt{10} = 13\sqrt{10}
\][/tex]
Therefore, the expression simplifies to [tex]\(13\sqrt{10}\)[/tex].
The correct choice is C: [tex]\(13\sqrt{10}\)[/tex].