To express [tex]\(\sqrt{11} + \sqrt{99}\)[/tex] in the form [tex]\(a \sqrt{b}\)[/tex], where [tex]\(a\)[/tex] and [tex]\(b\)[/tex] are integers, we can follow these steps:
1. Simplify [tex]\(\sqrt{99}\)[/tex]:
Notice that [tex]\(99\)[/tex] can be factored as [tex]\(9 \times 11\)[/tex]. Therefore, we can write:
[tex]\[
\sqrt{99} = \sqrt{9 \times 11}
\][/tex]
Using the property of square roots that [tex]\(\sqrt{a \times b} = \sqrt{a} \times \sqrt{b}\)[/tex], we get:
[tex]\[
\sqrt{99} = \sqrt{9} \times \sqrt{11}
\][/tex]
Since [tex]\(\sqrt{9} = 3\)[/tex], this further simplifies to:
[tex]\[
\sqrt{99} = 3\sqrt{11}
\][/tex]
2. Combine the terms involving [tex]\(\sqrt{11}\)[/tex]:
Now, we have:
[tex]\[
\sqrt{11} + \sqrt{99} = \sqrt{11} + 3\sqrt{11}
\][/tex]
Since both terms have a common factor of [tex]\(\sqrt{11}\)[/tex], we can combine them as follows:
[tex]\[
\sqrt{11} + 3\sqrt{11} = (1 + 3)\sqrt{11} = 4\sqrt{11}
\][/tex]
Therefore, the expression [tex]\(\sqrt{11} + \sqrt{99}\)[/tex] can be written in the form [tex]\(a \sqrt{b}\)[/tex] where [tex]\(a = 4\)[/tex] and [tex]\(b = 11\)[/tex].
