Sure, let's solve the expression [tex]\(10 \sqrt{11} - \sqrt{396} - 3 \sqrt{11}\)[/tex] step-by-step.
1. Identify the terms in the expression:
[tex]\[
10 \sqrt{11}, -\sqrt{396}, -3 \sqrt{11}
\][/tex]
2. Rewrite the middle term:
The middle term is [tex]\(\sqrt{396}\)[/tex]. Notice that [tex]\(396\)[/tex] can be factored into [tex]\(36 \times 11\)[/tex].
[tex]\[
\sqrt{396} = \sqrt{36 \times 11}
\][/tex]
Since [tex]\(\sqrt{36} \times \sqrt{11} = 6 \sqrt{11}\)[/tex], we can rewrite [tex]\(\sqrt{396}\)[/tex] as:
[tex]\[
\sqrt{396} = 6 \sqrt{11}
\][/tex]
3. Substitute [tex]\(\sqrt{396}\)[/tex] in the expression:
[tex]\[
10 \sqrt{11} - 6 \sqrt{11} - 3 \sqrt{11}
\][/tex]
4. Combine like terms:
All terms involve [tex]\(\sqrt{11}\)[/tex], so we can combine them:
[tex]\[
(10 \sqrt{11} - 6 \sqrt{11} - 3 \sqrt{11}) = (10 - 6 - 3) \sqrt{11} = 1 \sqrt{11}
\][/tex]
5. Simplified expression:
[tex]\[
\sqrt{11}
\][/tex]
However, by plugging in the calculated values:
- [tex]\(10 \sqrt{11} \approx 33.166247903553995\)[/tex]
- [tex]\(-6 \sqrt{11} \approx -19.8997487421324\)[/tex]
- [tex]\(-3 \sqrt{11} \approx -9.9498743710662\)[/tex]
- Add these values together, we get approximately [tex]\(3.3166247903553963\)[/tex].
Therefore, the final numerical result is:
[tex]\[
\boxed{3.3166247903553963}
\][/tex]
