To solve for the value of [tex]\(\sqrt{19 + 8 \sqrt{3}}\)[/tex], let's consider the given options and check which one, when squared, gives us the original expression.
The options are:
a. [tex]\(4 - \sqrt{3}\)[/tex]
b. [tex]\(4 + \sqrt{3}\)[/tex]
c. [tex]\(8 + \sqrt{3}\)[/tex]
d. [tex]\(8 - \sqrt{3}\)[/tex]
First, let's square each option to find if it matches [tex]\(19 + 8 \sqrt{3}\)[/tex].
1. Option [tex]\(a\)[/tex]: [tex]\((4 - \sqrt{3})^2\)[/tex]
[tex]\[
(4 - \sqrt{3})^2 = 4^2 - 2 \times 4 \times \sqrt{3} + (\sqrt{3})^2 = 16 - 8\sqrt{3} + 3 = 19 - 8\sqrt{3}
\][/tex]
This does not match [tex]\(19 + 8 \sqrt{3}\)[/tex].
2. Option [tex]\(b\)[/tex]: [tex]\((4 + \sqrt{3})^2\)[/tex]
[tex]\[
(4 + \sqrt{3})^2 = 4^2 + 2 \times 4 \times \sqrt{3} + (\sqrt{3})^2 = 16 + 8\sqrt{3} + 3 = 19 + 8\sqrt{3}
\][/tex]
This matches the original expression [tex]\(19 + 8 \sqrt{3}\)[/tex].
3. Option [tex]\(c\)[/tex]: [tex]\((8 + \sqrt{3})^2\)[/tex]
[tex]\[
(8 + \sqrt{3})^2 = 8^2 + 2 \times 8 \times \sqrt{3} + (\sqrt{3})^2 = 64 + 16\sqrt{3} + 3 = 67 + 16\sqrt{3}
\][/tex]
This does not match [tex]\(19 + 8 \sqrt{3}\)[/tex].
4. Option [tex]\(d\)[/tex]: [tex]\((8 - \sqrt{3})^2\)[/tex]
[tex]\[
(8 - \sqrt{3})^2 = 8^2 - 2 \times 8 \times \sqrt{3} + (\sqrt{3})^2 = 64 - 16\sqrt{3} + 3 = 67 - 16\sqrt{3}
\][/tex]
This does not match [tex]\(19 + 8 \sqrt{3}\)[/tex].
From the options, only option [tex]\(b\)[/tex], [tex]\((4 + \sqrt{3})^2\)[/tex], matches the original expression [tex]\(19 + 8 \sqrt{3}\)[/tex].
Thus, the value of [tex]\(\sqrt{19 + 8 \sqrt{3}}\)[/tex] is:
[tex]\[
4 + \sqrt{3}
\][/tex]
So, the correct answer is [tex]\(b\)[/tex].
