Let's solve the equation [tex]\(\sqrt{7+\sqrt{2+\sqrt[3]{a-5}}}=3\)[/tex] step-by-step.
1. Isolate the Outer Square Root:
Given [tex]\(\sqrt{7 + \sqrt{2 + \sqrt[3]{a - 5}}} = 3\)[/tex], to remove the outermost square root, square both sides of the equation:
[tex]\[
\left(\sqrt{7 + \sqrt{2 + \sqrt[3]{a - 5}}}\right)^2 = 3^2
\][/tex]
Simplifying this, we get:
[tex]\[
7 + \sqrt{2 + \sqrt[3]{a - 5}} = 9
\][/tex]
2. Isolate the Next Inner Square Root:
Now, move the 7 to the right side by subtracting 7 from both sides:
[tex]\[
\sqrt{2 + \sqrt[3]{a - 5}} = 9 - 7
\][/tex]
Simplifying this, we get:
[tex]\[
\sqrt{2 + \sqrt[3]{a - 5}} = 2
\][/tex]
3. Square Both Sides Again:
To remove the next square root, square both sides again:
[tex]\[
\left(\sqrt{2 + \sqrt[3]{a - 5}}\right)^2 = 2^2
\][/tex]
This simplifies to:
[tex]\[
2 + \sqrt[3]{a - 5} = 4
\][/tex]
4. Isolate the Cube Root:
Move the 2 to the right side by subtracting 2 from both sides:
[tex]\[
\sqrt[3]{a - 5} = 4 - 2
\][/tex]
Simplifying this, we get:
[tex]\[
\sqrt[3]{a - 5} = 2
\][/tex]
5. Solve for [tex]\(a\)[/tex]:
To remove the cube root, cube both sides of the equation:
[tex]\[
\left(\sqrt[3]{a - 5}\right)^3 = 2^3
\][/tex]
This simplifies to:
[tex]\[
a - 5 = 8
\][/tex]
6. Find the Value of [tex]\(a\)[/tex]:
Finally, solve for [tex]\(a\)[/tex] by adding 5 to both sides:
[tex]\[
a = 8 + 5
\][/tex]
This simplifies to:
[tex]\[
a = 13
\][/tex]
So, the solution to the equation [tex]\(\sqrt{7 + \sqrt{2 + \sqrt[3]{a - 5}}} = 3\)[/tex] is:
[tex]\[
a = 13
\][/tex]
