Linear Algebra Examples

$8 (\cos (\frac{π}{2}) + i \sin (\frac{π}{2}))$

Step 1

Calculate the distance from $(a, b)$ to the origin using the formula $r = \sqrt{a^{2} + b^{2}}$ .

$r = \sqrt{{(8 \cos (\frac{π}{2}))}^{2} + {(\sin (\frac{π}{2}) \cdot 8)}^{2}}$

Step 2

Simplify $\sqrt{{(8 \cos (\frac{π}{2}))}^{2} + {(\sin (\frac{π}{2}) \cdot 8)}^{2}}$ .

Tap for more steps...

Step 2.1

The exact value of $\cos (\frac{π}{2})$ is $0$ .

$r = \sqrt{{(8 \cdot 0)}^{2} + {(\sin (\frac{π}{2}) \cdot 8)}^{2}}$

Step 2.2

Multiply $8$ by $0$ .

$r = \sqrt{0^{2} + {(\sin (\frac{π}{2}) \cdot 8)}^{2}}$

Step 2.3

Raising $0$ to any positive power yields $0$ .

$r = \sqrt{0 + {(\sin (\frac{π}{2}) \cdot 8)}^{2}}$

Step 2.4

The exact value of $\sin (\frac{π}{2})$ is $1$ .

$r = \sqrt{0 + {(1 \cdot 8)}^{2}}$

Step 2.5

Multiply $8$ by $1$ .

$r = \sqrt{0 + 8^{2}}$

Step 2.6

Raise $8$ to the power of $2$ .

$r = \sqrt{0 + 64}$

Step 2.7

Add $0$ and $64$ .

$r = \sqrt{64}$

Step 2.8

Rewrite $64$ as $8^{2}$ .

$r = \sqrt{8^{2}}$

Step 2.9

Pull terms out from under the radical, assuming positive real numbers.

$r = 8$

Step 3

Calculate reference angle $θ̂ = \arctan (| \frac{b}{a} |)$ .

$θ̂ = \arctan (| \frac{\sin (\frac{π}{2}) \cdot 8}{8 \cos (\frac{π}{2})} |)$

Step 4

The equation has an undefined fraction.

Undefined

Step 5

Find the quadrant.

Tap for more steps...

Step 5.1

The exact value of $\cos (\frac{π}{2})$ is $0$ .

$(8 \cdot 0, \sin (\frac{π}{2}) \cdot 8)$

Step 5.2

Multiply $8$ by $0$ .

$(0, \sin (\frac{π}{2}) \cdot 8)$

Step 5.3

The exact value of $\sin (\frac{π}{2})$ is $1$ .

$(0, 1 \cdot 8)$

Step 5.4

Multiply $8$ by $1$ .

$(0, 8)$

Step 5.5

Since the y-coordinate is positive and the x-coordinate is $0$ , the point is located on y-axis between the first and fourth quadrants. The quadrants are labeled in counter-clockwise order, starting in the upper-right.

Between Quadrant $1$ and $2$

Step 6

Use the formula to find the roots of the complex number.

${(a + b i)}^{\frac{1}{n}} = r^{\frac{1}{n}} c i s (\frac{θ + 2 π k}{n})$ , $k = 0, 1, \dots, n - 1$

Step 7

Substitute $r$ , $n$ , and $θ$ into the formula.

Tap for more steps...

Step 7.1

Combine ${(8)}^{\frac{1}{3}}$ and $\frac{θ + 2 π k}{3}$ .

$c i s \frac{{(8)}^{\frac{1}{3}} (θ + 2 π k)}{3}$

Step 7.2

Combine $c$ and $\frac{{(8)}^{\frac{1}{3}} (θ + 2 π k)}{3}$ .

$i s \frac{c ({(8)}^{\frac{1}{3}} (θ + 2 π k))}{3}$

Step 7.3

Combine $i$ and $\frac{c ({(8)}^{\frac{1}{3}} (θ + 2 π k))}{3}$ .

$s \frac{i (c ({(8)}^{\frac{1}{3}} (θ + 2 π k)))}{3}$

Step 7.4

Combine $s$ and $\frac{i (c ({(8)}^{\frac{1}{3}} (θ + 2 π k)))}{3}$ .

$\frac{s (i (c ({(8)}^{\frac{1}{3}} (θ + 2 π k))))}{3}$

Step 7.5

Remove parentheses.

Tap for more steps...

Step 7.5.1

Remove parentheses.

$\frac{s (i (c (8^{\frac{1}{3}} (θ + 2 π k))))}{3}$

Step 7.5.2

Remove parentheses.

$\frac{s (i (c \cdot 8^{\frac{1}{3}} (θ + 2 π k)))}{3}$

Step 7.5.3

Remove parentheses.

$\frac{s (i (c \cdot 8^{\frac{1}{3}}) (θ + 2 π k))}{3}$

Step 7.5.4

Remove parentheses.

$\frac{s (i c \cdot 8^{\frac{1}{3}} (θ + 2 π k))}{3}$

Step 7.5.5

Remove parentheses.

$\frac{s (i c \cdot 8^{\frac{1}{3}}) (θ + 2 π k)}{3}$

Step 7.5.6

Remove parentheses.

$\frac{s (i c) \cdot 8^{\frac{1}{3}} (θ + 2 π k)}{3}$

Step 7.5.7

Remove parentheses.

$\frac{s i c \cdot 8^{\frac{1}{3}} (θ + 2 π k)}{3}$

Step 8

Substitute $k = 0$ into the formula and simplify.

Tap for more steps...

Step 8.1

Rewrite $8$ as $2^{3}$ .

$k = 0 : {(2^{3})}^{\frac{1}{3}} c i s (\frac{θ + 2 π (0)}{3})$

Step 8.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$k = 0 : 2^{3 (\frac{1}{3})} c i s (\frac{θ + 2 π (0)}{3})$

Step 8.3

Cancel the common factor of $3$ .

Tap for more steps...

Step 8.3.1

Cancel the common factor.

$k = 0 : 2^{3 (\frac{1}{3})} c i s (\frac{θ + 2 π (0)}{3})$

Step 8.3.2

Rewrite the expression.

$k = 0 : 2 c i s (\frac{θ + 2 π (0)}{3})$

Step 8.4

Evaluate the exponent.

$k = 0 : 2 c i s (\frac{θ + 2 π (0)}{3})$

Step 8.5

Multiply $2 π (0)$ .

Tap for more steps...

Step 8.5.1

Multiply $0$ by $2$ .

$k = 0 : 2 c i s (\frac{θ + 0 π}{3})$

Step 8.5.2

Multiply $0$ by $π$ .

$k = 0 : 2 c i s (\frac{θ + 0}{3})$

Step 9

Substitute $k = 1$ into the formula and simplify.

Tap for more steps...

Step 9.1

Rewrite $8$ as $2^{3}$ .

$k = 1 : {(2^{3})}^{\frac{1}{3}} c i s (\frac{θ + 2 π (1)}{3})$

Step 9.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$k = 1 : 2^{3 (\frac{1}{3})} c i s (\frac{θ + 2 π (1)}{3})$

Step 9.3

Cancel the common factor of $3$ .

Tap for more steps...

Step 9.3.1

Cancel the common factor.

$k = 1 : 2^{3 (\frac{1}{3})} c i s (\frac{θ + 2 π (1)}{3})$

Step 9.3.2

Rewrite the expression.

$k = 1 : 2 c i s (\frac{θ + 2 π (1)}{3})$

Step 9.4

Evaluate the exponent.

$k = 1 : 2 c i s (\frac{θ + 2 π (1)}{3})$

Step 9.5

Multiply $2$ by $1$ .

$k = 1 : 2 c i s (\frac{θ + 2 π}{3})$

Step 10

Substitute $k = 2$ into the formula and simplify.

Tap for more steps...

Step 10.1

Rewrite $8$ as $2^{3}$ .

$k = 2 : {(2^{3})}^{\frac{1}{3}} c i s (\frac{θ + 2 π (2)}{3})$

Step 10.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$k = 2 : 2^{3 (\frac{1}{3})} c i s (\frac{θ + 2 π (2)}{3})$

Step 10.3

Cancel the common factor of $3$ .

Tap for more steps...

Step 10.3.1

Cancel the common factor.

$k = 2 : 2^{3 (\frac{1}{3})} c i s (\frac{θ + 2 π (2)}{3})$

Step 10.3.2

Rewrite the expression.

$k = 2 : 2 c i s (\frac{θ + 2 π (2)}{3})$

Step 10.4

Evaluate the exponent.

$k = 2 : 2 c i s (\frac{θ + 2 π (2)}{3})$

Step 10.5

Multiply $2$ by $2$ .

$k = 2 : 2 c i s (\frac{θ + 4 π}{3})$

Step 11

List the solutions.

$k = 0 : 2 c i s (\frac{θ + 0}{3})$

$k = 1 : 2 c i s (\frac{θ + 2 π}{3})$

$k = 2 : 2 c i s (\frac{θ + 4 π}{3})$