To find the value of [tex]\( x^3 + \frac{1}{x^3} \)[/tex] given that [tex]\( x - \frac{1}{x} = \frac{1}{6} \)[/tex], we can use some algebraic identities and manipulation.
1. Let's denote [tex]\( y \)[/tex] as [tex]\( x - \frac{1}{x} \)[/tex]:
[tex]\[
y = \frac{1}{6}
\][/tex]
2. We need to find [tex]\( x^3 + \frac{1}{x^3} \)[/tex]. To do this, we use the identity:
[tex]\[
(x - \frac{1}{x})^3 = x^3 - 3x + 3 \cdot \frac{1}{x} - \frac{1}{x^3}
\][/tex]
Simplifying, we get:
[tex]\[
(x - \frac{1}{x})^3 = x^3 + \frac{1}{x^3} - 3(x - \frac{1}{x})
\][/tex]
3. Substituting [tex]\( y \)[/tex] and [tex]\( y^3 \)[/tex] into this identity, we obtain:
[tex]\[
y^3 = x^3 + \frac{1}{x^3} - 3y
\][/tex]
4. Since [tex]\( y = \frac{1}{6} \)[/tex], we can calculate [tex]\( y^3 \)[/tex]:
[tex]\[
y^3 = \left( \frac{1}{6} \right)^3 = \frac{1}{216}
\][/tex]
5. Now, substitute [tex]\( y \)[/tex] and [tex]\( y^3 \)[/tex] back into the equation:
[tex]\[
\frac{1}{216} = x^3 + \frac{1}{x^3} - 3 \cdot \frac{1}{6}
\][/tex]
6. Calculate [tex]\( 3 \cdot \frac{1}{6} \)[/tex]:
[tex]\[
3 \cdot \frac{1}{6} = \frac{1}{2}
\][/tex]
7. Rearrange the equation to solve for [tex]\( x^3 + \frac{1}{x^3} \)[/tex]:
[tex]\[
\frac{1}{216} + \frac{1}{2} = x^3 + \frac{1}{x^3}
\][/tex]
8. Simplify the right-hand side:
[tex]\[
\frac{1}{2} = 0.5 \quad \text{and} \quad \frac{1}{216} = 0.00462962962962963
\][/tex]
9. Adding these values together:
[tex]\[
x^3 + \frac{1}{x^3} = 0.5 + 0.00462962962962963 = 0.5046296296296297
\][/tex]
Thus, the value of [tex]\( x^3 + \frac{1}{x^3} \)[/tex] is:
[tex]\[
\boxed{0.5046296296296297}
\][/tex]
