Raise the number to the given power and write the answer in rectangular form.

Simplify your answer, including any radicals. Use integers or fractions for any numbers in the

Ask by Bowman Gray. in the United States

Nov 22,2024

Question

Raise the number to the given power and write the answer in rectangular form.

Simplify your answer, including any radicals. Use integers or fractions for any numbers in the

Ask by Bowman Gray. in the United States

Nov 22,2024

UpStudy AI · Accepted Answer

To raise the number $$ 2 \left( \text{cis} 105^\circ \right) $$ to the power of 3, we can use the properties of complex numbers in polar form. The expression $$ \text{cis} \theta $$ is shorthand for $$ \cos \theta + i \sin \theta $$.

1. **Apply the power to the modulus and the angle**:
   $$
   \left[2 \left( \text{cis} 105^\circ\right)\right]^3 = 2^3 \left( \text{cis} (3 \times 105^\circ) \right)
   $$

2. **Calculate $$ 2^3 $$**:
   $$
   2^3 = 8
   $$

3. **Calculate the new angle**:
   $$
   3 \times 105^\circ = 315^\circ
   $$

4. **Combine the results**:
   $$
   \left[2 \left( \text{cis} 105^\circ\right)\right]^3 = 8 \left( \text{cis} 315^\circ \right)
   $$

5. **Convert $$ \text{cis} 315^\circ $$ to rectangular form**:
   $$
   \text{cis} 315^\circ = \cos 315^\circ + i \sin 315^\circ
   $$
   The values of $$ \cos 315^\circ $$ and $$ \sin 315^\circ $$ are:
   $$
   \cos 315^\circ = \frac{\sqrt{2}}{2}, \quad \sin 315^\circ = -\frac{\sqrt{2}}{2}
   $$
   Therefore,
   $$
   \text{cis} 315^\circ = \frac{\sqrt{2}}{2} - i \frac{\sqrt{2}}{2}
   $$

6. **Multiply by 8**:
   $$
   8 \left( \text{cis} 315^\circ \right) = 8 \left( \frac{\sqrt{2}}{2} - i \frac{\sqrt{2}}{2} \right) = 8 \cdot \frac{\sqrt{2}}{2} - 8 \cdot i \cdot \frac{\sqrt{2}}{2}
   $$
   Simplifying this gives:
   $$
   4\sqrt{2} - 4\sqrt{2} i
   $$

Thus, the final answer in rectangular form is:
$$
\left[2\left( \text{cis} 105^{\circ}\right)\right]^{3} = 4\sqrt{2} - 4\sqrt{2} i
$$
$$ \left[2\left( \text{cis} 105^{\circ}\right)\right]^{3} = 4\sqrt{2} - 4\sqrt{2} i $$

Raise the number to the given power and write the answer in rectangular form.

Simplify your answer, including any radicals. Use integers or fractions for any numbers in the

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