Let's solve the given problem step-by-step.
We start with the given equation:
[tex]\[
\sqrt{7 + \sqrt{48}} = m + \sqrt{n}
\][/tex]
First, simplify the expression inside the square root. Notice that:
[tex]\[
\sqrt{48} = \sqrt{16 \cdot 3} = \sqrt{16} \cdot \sqrt{3} = 4\sqrt{3}
\][/tex]
So the equation now becomes:
[tex]\[
\sqrt{7 + 4\sqrt{3}} = m + \sqrt{n}
\][/tex]
Assume [tex]\( m = 2 \)[/tex] and [tex]\( n = 3 \)[/tex]. We need to verify this assumption by seeing if squaring both sides gives us a correct equality. Thus,
[tex]\[
(m + \sqrt{n})^2 = (2 + \sqrt{3})^2
\][/tex]
Calculate the square on the right-hand side:
[tex]\[
(2 + \sqrt{3})^2 = 2^2 + 2 \cdot 2 \cdot \sqrt{3} + (\sqrt{3})^2
\][/tex]
Breaking it down, we get:
[tex]\[
4 + 4\sqrt{3} + 3 = 7 + 4\sqrt{3}
\][/tex]
So, our assumptions of [tex]\( m = 2 \)[/tex] and [tex]\( n = 3 \)[/tex] are correct because the expressions match:
[tex]\[
\sqrt{7 + 4\sqrt{3}} = 2 + \sqrt{3}
\][/tex]
Now, we need to find [tex]\( m^2 + n^2 \)[/tex].
Calculate [tex]\( m^2 \)[/tex]:
[tex]\[
m^2 = 2^2 = 4
\][/tex]
Calculate [tex]\( n^2 \)[/tex]:
[tex]\[
n^2 = 3^2 = 9
\][/tex]
Then, add these results together:
[tex]\[
m^2 + n^2 = 4 + 9 = 13
\][/tex]
Thus, the value of [tex]\( m^2 + n^2 \)[/tex] is:
[tex]\[
\boxed{13}
\][/tex]
