To solve the equation [tex]\(\sqrt{7+\sqrt{2+\sqrt[3]{a-5}}}=3\)[/tex], let's go through a detailed, step-by-step process:
1. Square both sides of the equation to eliminate the outer square root:
[tex]\[
\left( \sqrt{7 + \sqrt{2 + \sqrt[3]{a-5}}} \right)^2 = 3^2
\][/tex]
This simplifies to:
[tex]\[
7 + \sqrt{2 + \sqrt[3]{a-5}} = 9
\][/tex]
2. Isolate the inner square root by subtracting 7 from both sides of the equation:
[tex]\[
\sqrt{2 + \sqrt[3]{a-5}} = 9 - 7
\][/tex]
This simplifies to:
[tex]\[
\sqrt{2 + \sqrt[3]{a-5}} = 2
\][/tex]
3. Square both sides again to eliminate the second square root:
[tex]\[
\left( \sqrt{2 + \sqrt[3]{a-5}} \right)^2 = 2^2
\][/tex]
Simplifying, we get:
[tex]\[
2 + \sqrt[3]{a-5} = 4
\][/tex]
4. Isolate the cube root term by subtracting 2 from both sides:
[tex]\[
\sqrt[3]{a-5} = 4 - 2
\][/tex]
This simplifies to:
[tex]\[
\sqrt[3]{a-5} = 2
\][/tex]
5. Cube both sides to eliminate the cube root:
[tex]\[
\left( \sqrt[3]{a-5} \right)^3 = 2^3
\][/tex]
Simplifying, we get:
[tex]\[
a - 5 = 8
\][/tex]
6. Solve for [tex]\(a\)[/tex] by adding 5 to both sides:
[tex]\[
a = 8 + 5
\][/tex]
Thus:
[tex]\[
a = 13
\][/tex]
Therefore, the value of [tex]\(a\)[/tex] that satisfies the given equation is [tex]\(\boxed{13}\)[/tex].
