Let's solve the expression [tex]\(\left(-\frac{3}{4}\right)^{\frac{2}{3}}\)[/tex] step-by-step.
To deal with the fractional exponent of a negative base, we need to consider complex numbers because raising a negative number to a fractional power generally results in a complex number.
Given:
[tex]\[
\left( -\frac{3}{4} \right)^{\frac{2}{3}}
\][/tex]
First, let's convert [tex]\(-\frac{3}{4}\)[/tex] into its polar form. The polar form of a complex number [tex]\(r e^{i \theta}\)[/tex] can be useful in evaluating exponents. Here, [tex]\(r\)[/tex] is the magnitude and [tex]\(\theta\)[/tex] is the argument (angle).
1. Calculate the magnitude:
[tex]\[ r = \left| -\frac{3}{4} \right| = \frac{3}{4} \][/tex]
2. Determine the argument:
[tex]\[ \theta = \pi \quad \text{(since the number is negative, its angle with the positive real axis is } \pi \text{ radians)} \][/tex]
Now, express [tex]\(-\frac{3}{4}\)[/tex] in its polar form:
[tex]\[ -\frac{3}{4} = \frac{3}{4} e^{i \pi} \][/tex]
Now we need to raise this to the power of [tex]\(\frac{2}{3}\)[/tex]:
[tex]\[
\left( \frac{3}{4} e^{i \pi} \right)^{\frac{2}{3}}
\][/tex]
Using properties of exponents in polar form:
[tex]\[
\left( \frac{3}{4} \right)^{\frac{2}{3}} e^{i \pi \cdot \frac{2}{3}}
\][/tex]
3. Calculate the magnitude part:
[tex]\[
\left( \frac{3}{4} \right)^{\frac{2}{3}}
\][/tex]
4. Compute the argument part:
[tex]\[
e^{i \cdot \frac{2}{3} \pi}
\][/tex]
Putting it all together:
[tex]\[
\left( \frac{3}{4} \right)^{\frac{2}{3}} \cdot e^{i \cdot \frac{2}{3} \pi}
\][/tex]
Evaluate the magnitude:
[tex]\[
\left( \frac{3}{4} \right)^{\frac{2}{3}} \approx 0.641
\][/tex]
Evaluate the argument:
[tex]\[
e^{i \cdot \frac{2}{3} \pi} = \cos \left( \frac{2}{3} \pi \right) + i \sin \left( \frac{2}{3} \pi \right)
\][/tex]
Using the values of [tex]\(\cos \left( \frac{2}{3} \pi \right)\)[/tex] and [tex]\(\sin \left( \frac{2}{3} \pi \right)\)[/tex]:
[tex]\[
\cos \left( \frac{2}{3} \pi \right) \approx -0.5 \quad \text{and} \quad \sin \left( \frac{2}{3} \pi \right) \approx 0.866
\][/tex]
Thus:
[tex]\[
e^{i \cdot \frac{2}{3} \pi} \approx -0.5 + 0.866i
\][/tex]
Combining both parts:
[tex]\[
\left( \frac{3}{4} \right)^{\frac{2}{3}} \cdot \left( -0.5 + 0.866i \right)
\][/tex]
The final result is:
[tex]\[
(-0.4127409061118282 + 0.7148882197477024i)
\][/tex]
So, the result of [tex]\(\left( -\frac{3}{4} \right)^{\frac{2}{3}}\)[/tex] is:
[tex]\[
\boxed{(-0.4127409061118282 + 0.7148882197477024i)}
\][/tex]
