To simplify [tex]\((-729)^{\frac{1}{3}}\)[/tex], we need to find the cube root of [tex]\(-729\)[/tex]. Complex numbers play a role in this, as negative real numbers have cube roots that are complex.
1. Express [tex]\(-729\)[/tex] as [tex]\( -1 \times 729 \)[/tex].
2. We can separate these terms in the context of cube roots:
[tex]\[
(-729)^{\frac{1}{3}} = (-1 \times 729)^{\frac{1}{3}} = (-1)^{\frac{1}{3}} \times 729^{\frac{1}{3}}
\][/tex]
3. The cube root of [tex]\(729\)[/tex]:
[tex]\[
729^{\frac{1}{3}} = 9 \, \text{(since \(9^3 = 729\))}
\][/tex]
4. Next, we need to find the cube root of [tex]\(-1\)[/tex]. In complex number theory, the cube roots of [tex]\(-1\)[/tex] are complex numbers, and one of them is typically expressed as:
[tex]\[
(-1)^{\frac{1}{3}} = e^{i\pi/3} = \frac{1}{2} + \frac{\sqrt{3}}{2} i
\][/tex]
However, for a complete set of roots, it's crucial to consider the principal value and the nth roots of unity multiplication.
5. Combining these results, we get:
[tex]\[
(-1)^{\frac{1}{3}} \approx 0.5 + 0.866i \,(\text{approximation of a cube root of -1})
\][/tex]
6. Multiplying this result by [tex]\(9\)[/tex]:
[tex]\[
9 (0.5 + 0.866i) = 4.5 + 7.794i
\][/tex]
Therefore, the simplified form of
[tex]\[
(-729)^{\frac{1}{3}}
\][/tex]
results in:
[tex]\[
4.5 + 7.794i
\][/tex]
So, the solution is:
[tex]\[
(-729)^{\frac{1}{3}} = 4.5 + 7.794i
\][/tex]
