Question

Using $\mathit{a}\mathbf{=}\mathbf{1}\mathbf{\angle }\mathbf{120}\mathbf{°}\mathbf{,}$$\mathit{a}\mathbf{=}\mathbf{\angle }\mathbf{120}\mathbf{°}$,evaluate the following in rectangular form:

a. role="math" localid="1655411384516" ${\mathit{a}}^{\mathbf{10}}$localid="1654583442123" ${\mathit{a}}^{\mathbf{10}}$

b. ${\left(ja\right)}^{\mathbf{10}}$ localid="1654583450911" ${\mathbf{\left(}\mathbf{j}\mathbf{a}\mathbf{\right)}}^{\mathbf{10}}$

c. ${\left(1-a\right)}^{\mathbf{3}}$localid="1654583456163" ${\mathbf{(}\mathbf{1}\mathbf{-}\mathbf{a}\mathbf{)}}^{\mathbf{3}}$

d. role="math" localid="1655411466773" ${\mathit{e}}^{\mathbf{a}}$localid="1654583461053" ${\mathit{e}}^{\mathbf{a}}$

Short answer

a) After evaluating the expression: $a^{10}$the value obtained in rectangular form is $-\frac{1}{2}+\frac{\sqrt{3}}{2}j$$-\frac{1}{2}+\frac{\sqrt{3}}{2}j$

b) After evaluating the expression: ${\left(ja\right)}^{10}$the value obtained in rectangular form is $\frac{1}{2}-\frac{\sqrt{3}}{2}j$$\frac{1}{2}-\frac{\sqrt{3}}{2}j$

c) After evaluating the expression:${\left(1-a\right)}^3$the value obtained in rectangular form is$-5.19j$$-5.19j$

d) Therefore, after evaluating the expression: $e^a$, the value obtained in rectangular form is$0.3929+j0.4620$$0.3929+j0.4620$

Step-by-step solution

 Express a10 in the rectangular form

a)

Determine the value of the expression:$a^{10}$$a^{10}$

As given in the data, substitute the value $1\angle 120°$for $a$in the above expression.

$$\begin{gathered} a^{10}={\left(a^3\right)}^3\times a^1 \\ ={\left({\left(1\angle 120°\right)}^3\right)}^3\times {\left(1\angle 120°\right)}^1 \\ ={\left(\left(1\angle 360°\right)\right)}^3\times {\left(1\angle 120°\right)}^1 \\ ={\left(1\angle 0°\right)}^3\times \left(1\angle 120°\right) \end{gathered}$$

Further solving,

$$\begin{gathered} a^{10}=\left(1\right)\left(1\angle 120°\right) \\ =\left(1\angle 120°\right) \end{gathered}$$

Now convert the above polar expression into the rectangular form;

$a^{10}=\frac{1}{2}+\frac{\sqrt{3}}{2}j$

Therefore, after evaluating the expression: , the value obtained in rectangular form is$-\frac{1}{2}+\frac{\sqrt{3}}{2}j$

Express ja10 in the rectangular form

(b)

Determine the value of the expression:

As given in the data, substitute the value $-\frac{1}{2}+\frac{\sqrt{3}}{2}j$for $a^{10}$in the above expression.

$$\begin{gathered} {\left(ja\right)}^{10}={\left[j\left(a\right)\right]}^{10} \\ ={\left(j\right)}^{10}\left(a^{10}\right) \\ ={\left(j\right)}^2{\left(j\right)}^4{\left(j\right)}^4\left(a^{10}\right) \\ =\left(-1\right)\left(1\right)\left(1\right)\left(-\frac{1}{2}+\frac{\sqrt{3}}{2}j\right) \end{gathered}$$

Further solving,

$$\begin{gathered} {\left(j\right)}^{10}=-\left(-\frac{1}{2}+\frac{\sqrt{3}}{2}j\right) \\ =\frac{1}{2}-\frac{\sqrt{3}}{2}j \end{gathered}$$

Therefore, after evaluating the expression:${\left(ja\right)}^{10}$, the value obtained in rectangular form is $\frac{1}{2}-\frac{\sqrt{3}}{2}j$.

Express 1-a3 in the rectangular form

(c)

Determine the value of the expression:${\left(1-a\right)}^3$

As given in the data, substitute the value $1\angle 120°$for $a$in the above expression.

${\left(1-a\right)}^3={\left[1-\left(1\angle 120°\right)\right]}^3$

Convert the above given expression into rectangular form;

$$\begin{gathered} {\left(1-a\right)}^3={\left[1-\left(-\frac{1}{2}+\frac{\sqrt{3}}{2}j\right)\right]}^3 \\ ={\left[1+\frac{1}{2}+\frac{\sqrt{3}}{2}j\right]}^3 \\ ={\left[\frac{3}{2}-\frac{\sqrt{3}}{2}j\right]}^3 \\ =3.37-5.48j-3.37j^2-0.65j^3 \end{gathered}$$

Further solving,

$$\begin{gathered} {\left(1-a\right)}^3=3.37-5.84j-3.37+0.65j \\ =-5.19j \end{gathered}$$

Therefore, after evaluating the expression: ${\left(1-a\right)}^3$, the value obtained in rectangular form is .$-5.19j$

Express ea in the rectangular form

(d)

Determine the value of the expression:$e^a$

As given in the data, substitute the value $1\angle 120°$for $a$and $1\angle 0redians$for $e^{j0}$in the above expression.

$e^a=e^{\left(1\angle 120°\right)}$

Convert the above given expression into rectangular form;

$$\begin{gathered} e^a=e^{\left(-\frac{1}{2}+\frac{\sqrt{3}}{2}j\right)} \\ =\left(e^{-\frac{1}{2}}\right)\left(e^{\frac{\sqrt{3}}{2}j}\right) \\ =\left(e^{-\frac{1}{2}}\right)\left(1\angle \left(\frac{\sqrt{3}}{2}\right)radians\right) \\ =\left(0.6065\right)\left(1\angle \left(\frac{0.866\times 180°}{\mathrm{\pi }}\right)\right) \end{gathered}$$

Further solving,

$$\begin{gathered} e^a=0.6065\angle 49.62° \\ =0.3929+j0.4620 \end{gathered}$$

Therefore, after evaluating the expression: $e^a$, the value obtained in rectangular form is $0.3929+j0.4620$.