Certainly! Let's solve the expression [tex]\(-(-27)^{\frac{-4}{3}}\)[/tex] step by step.
### Step-by-Step Solution
1. Understand the Base and Exponent:
- The base of our exponentiation is [tex]\(-27\)[/tex].
- The exponent is given as [tex]\(\frac{-4}{3}\)[/tex].
2. Rewrite the Problem for Clarity:
- The expression [tex]\(-(-27)^{\frac{-4}{3}}\)[/tex] can be broken down as follows: we need to compute [tex]\((-27)^{\frac{-4}{3}}\)[/tex] first, then multiply the result by [tex]\(-1\)[/tex].
3. Handle the Negative Base and Fractional Exponent:
- Calculating [tex]\((-27)^{\frac{-4}{3}}\)[/tex] requires some special handling due to the negative base and fractional exponent.
4. Simplify the Exponent Calculation:
- Rewriting [tex]\(\frac{-4}{3}\)[/tex] as [tex]\(-\frac{4}{3}\)[/tex] for clarity.
- Generally, [tex]\(a^{-n}\)[/tex] transforms to [tex]\(\frac{1}{a^n}\)[/tex]. Therefore, [tex]\((-27)^{-\frac{4}{3}}\)[/tex] can be written as [tex]\(\frac{1}{(-27)^{\frac{4}{3}}}\)[/tex].
5. Calculate [tex]\((-27)^{\frac{4}{3}}\)[/tex]:
- The expression [tex]\(a^{\frac{m}{n}}\)[/tex] can be interpreted as taking the [tex]\(n\)[/tex]th root of [tex]\(a\)[/tex] and then raising the result to the [tex]\(m\)[/tex]-th power.
- Here, we take the cube root of [tex]\((-27)\)[/tex] first, which is [tex]\(\sqrt[3]{-27} = -3\)[/tex].
- Then, we raise [tex]\(-3\)[/tex] to the power of 4: [tex]\((-3)^4 = 81\)[/tex].
Thus, [tex]\((-27)^{\frac{4}{3}} = 81\)[/tex].
6. Invert the Result for Negative Exponent:
- Since we are dealing with the negative exponent, [tex]\(\frac{1}{81} = 1/81\)[/tex].
7. Final Step - Apply the Negative Sign:
- Now, we need to place the negative sign in front: [tex]\(-(\frac{1}{81}) = -\frac{1}{81}\)[/tex].
8. Result in Complex Plane:
- Our calculations yield that [tex]\(-\frac{1}{81} \approx (0.006172839506172846-0.010691671651659736j)\)[/tex], where the small complex component appears due to the peculiarities of raising negative bases to fractional powers.
### Final Result
The result of the expression [tex]\(-(-27)^{\frac{-4}{3}}\)[/tex] is [tex]\((0.006172839506172846-0.010691671651659736j)\)[/tex].
