Welcome to Westonci.ca, your ultimate destination for finding answers to a wide range of questions from experts. Join our platform to connect with experts ready to provide precise answers to your questions in various areas. Explore comprehensive solutions to your questions from a wide range of professionals on our user-friendly platform.
Sagot :
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]
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]
We hope our answers were helpful. Return anytime for more information and answers to any other questions you may have. We hope this was helpful. Please come back whenever you need more information or answers to your queries. Get the answers you need at Westonci.ca. Stay informed by returning for our latest expert advice.
