Let's solve the equation step by step to find [tex]\(64x^3 + \frac{1}{64x^3}\)[/tex] given the initial equation [tex]\(4x + \frac{1}{4x} = 2\)[/tex].
1. Start with the given equation:
[tex]\[
4x + \frac{1}{4x} = 2
\][/tex]
2. Set a substitution for simplicity:
Let [tex]\( y = 4x \)[/tex]. Then the equation becomes:
[tex]\[
y + \frac{1}{y} = 2
\][/tex]
3. Multiply both sides by [tex]\( y \)[/tex] to eliminate the fraction:
[tex]\[
y^2 + 1 = 2y
\][/tex]
Rearrange it to the standard quadratic form:
[tex]\[
y^2 - 2y + 1 = 0
\][/tex]
4. Solve the quadratic equation:
[tex]\[
(y - 1)^2 = 0
\][/tex]
Taking the square root of both sides:
[tex]\[
y - 1 = 0 \implies y = 1
\][/tex]
5. Back-substitute [tex]\( y \)[/tex]:
Recalling that [tex]\( y = 4x \)[/tex], we have:
[tex]\[
4x = 1 \implies x = \frac{1}{4}
\][/tex]
6. Compute [tex]\( 64x^3 + \frac{1}{64x^3} \)[/tex]:
First, find [tex]\( x^3 \)[/tex]:
[tex]\[
x = \frac{1}{4} \implies x^3 = \left( \frac{1}{4} \right)^3 = \frac{1}{64}
\][/tex]
Next, find [tex]\( \frac{1}{64x^3} \)[/tex]:
[tex]\[
\frac{1}{64x^3} = \frac{1}{64 \left( \frac{1}{64} \right)} = 64
\][/tex]
Therefore:
[tex]\[
64x^3 + \frac{1}{64x^3} = 64 \left( \frac{1}{64} \right) + 64 = 1 + 64 = 65
\][/tex]
Thus, we find:
[tex]\[
64x^3 + \frac{1}{64x^3} = 2
\][/tex]
Therefore, the value of [tex]\(64x^3 + \frac{1}{64x^3}\)[/tex] is [tex]\(\boxed{2}\)[/tex].
