To solve the equation [tex]\( \sqrt[4]{x-4} = 3 \)[/tex], we need to follow several steps to isolate [tex]\( x \)[/tex].
Step 1: Eliminate the fourth root.
To remove the fourth root, we raise both sides of the equation to the power of 4. This step is important because raising a fourth root to the fourth power will cancel the root:
[tex]\[
(\sqrt[4]{x-4})^4 = 3^4
\][/tex]
Step 2: Simplify both sides.
The left side simplifies to [tex]\( x-4 \)[/tex], and the right side simplifies to [tex]\( 3^4 \)[/tex]:
[tex]\[
x-4 = 3^4
\][/tex]
Calculating [tex]\( 3^4 \)[/tex]:
[tex]\[
3^4 = 3 \times 3 \times 3 \times 3 = 81
\][/tex]
Thus, the equation becomes:
[tex]\[
x - 4 = 81
\][/tex]
Step 3: Solve for [tex]\( x \)[/tex].
Now, add 4 to both sides of the equation to isolate [tex]\( x \)[/tex]:
[tex]\[
x = 81 + 4
\][/tex]
[tex]\[
x = 85
\][/tex]
Therefore, the solution to the equation [tex]\( \sqrt[4]{x-4} = 3 \)[/tex] is [tex]\( x \)[/tex].
The correct answer is:
C. 85
