Let's break down and simplify the given expression [tex]\(2 \sqrt{54} + 5 \sqrt{24}\)[/tex].
First, let's simplify [tex]\(\sqrt{54}\)[/tex]:
1. Notice that 54 can be factored into 9 and 6:
[tex]\[
54 = 9 \times 6
\][/tex]
2. So,
[tex]\[
\sqrt{54} = \sqrt{9 \times 6} = \sqrt{9} \times \sqrt{6} = 3 \sqrt{6}
\][/tex]
Next, simplify [tex]\(\sqrt{24}\)[/tex]:
1. Notice that 24 can be factored into 4 and 6:
[tex]\[
24 = 4 \times 6
\][/tex]
2. So,
[tex]\[
\sqrt{24} = \sqrt{4 \times 6} = \sqrt{4} \times \sqrt{6} = 2 \sqrt{6}
\][/tex]
Now, substitute these simplified forms back into the original expression:
1. Replace [tex]\(\sqrt{54}\)[/tex] with [tex]\(3 \sqrt{6}\)[/tex]:
[tex]\[
2 \sqrt{54} = 2 \times 3 \sqrt{6} = 6 \sqrt{6}
\][/tex]
2. Replace [tex]\(\sqrt{24}\)[/tex] with [tex]\(2 \sqrt{6}\)[/tex]:
[tex]\[
5 \sqrt{24} = 5 \times 2 \sqrt{6} = 10 \sqrt{6}
\][/tex]
Now, add the two simplified terms together:
[tex]\[
6 \sqrt{6} + 10 \sqrt{6}
\][/tex]
Since both terms have the common radical [tex]\(\sqrt{6}\)[/tex], we can combine them:
[tex]\[
6 \sqrt{6} + 10 \sqrt{6} = (6 + 10) \sqrt{6} = 16 \sqrt{6}
\][/tex]
The expression [tex]\(2 \sqrt{54} + 5 \sqrt{24}\)[/tex] simplifies to:
[tex]\[
\boxed{16 \sqrt{6}}
\][/tex]
