Linear Algebra Examples

Find the Fourth Roots of a Complex Number -4-4i
Step 1
Calculate the distance from to the origin using the formula .
Step 2
Simplify .
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Step 2.1
Raise to the power of .
Step 2.2
Raise to the power of .
Step 2.3
Add and .
Step 2.4
Rewrite as .
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Step 2.4.1
Factor out of .
Step 2.4.2
Rewrite as .
Step 2.5
Pull terms out from under the radical.
Step 3
Calculate reference angle .
Step 4
Simplify .
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Step 4.1
Divide by .
Step 4.2
The absolute value is the distance between a number and zero. The distance between and is .
Step 4.3
The exact value of is .
Step 5
The point is located in the third quadrant because and are both negative. The quadrants are labeled in counter-clockwise order, starting in the upper-right.
Quadrant
Step 6
is in the third quadrant.
Step 7
Simplify .
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Step 7.1
To write as a fraction with a common denominator, multiply by .
Step 7.2
Combine fractions.
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Step 7.2.1
Combine and .
Step 7.2.2
Combine the numerators over the common denominator.
Step 7.3
Simplify the numerator.
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Step 7.3.1
Move to the left of .
Step 7.3.2
Add and .
Step 8
Use the formula to find the roots of the complex number.
,
Step 9
Substitute , , and into the formula.
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Step 9.1
To write as a fraction with a common denominator, multiply by .
Step 9.2
Combine and .
Step 9.3
Combine the numerators over the common denominator.
Step 9.4
Add and .
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Step 9.4.1
Reorder and .
Step 9.4.2
Add and .
Step 9.5
Combine and .
Step 9.6
Combine and .
Step 9.7
Combine and .
Step 9.8
Combine and .
Step 9.9
Remove parentheses.
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Step 9.9.1
Remove parentheses.
Step 9.9.2
Remove parentheses.
Step 9.9.3
Remove parentheses.
Step 9.9.4
Remove parentheses.
Step 9.9.5
Remove parentheses.
Step 9.9.6
Remove parentheses.
Step 10
Substitute into the formula and simplify.
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Step 10.1
Apply the product rule to .
Step 10.2
To write as a fraction with a common denominator, multiply by .
Step 10.3
Combine and .
Step 10.4
Combine the numerators over the common denominator.
Step 10.5
Simplify the numerator.
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Step 10.5.1
Move to the left of .
Step 10.5.2
Add and .
Step 10.6
Multiply .
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Step 10.6.1
Multiply by .
Step 10.6.2
Multiply by .
Step 10.7
Add and .
Step 10.8
Multiply the numerator by the reciprocal of the denominator.
Step 10.9
Multiply .
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Step 10.9.1
Multiply by .
Step 10.9.2
Multiply by .
Step 11
Substitute into the formula and simplify.
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Step 11.1
Apply the product rule to .
Step 11.2
To write as a fraction with a common denominator, multiply by .
Step 11.3
Combine and .
Step 11.4
Combine the numerators over the common denominator.
Step 11.5
Simplify the numerator.
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Step 11.5.1
Move to the left of .
Step 11.5.2
Add and .
Step 11.6
Multiply by .
Step 11.7
To write as a fraction with a common denominator, multiply by .
Step 11.8
Combine and .
Step 11.9
Combine the numerators over the common denominator.
Step 11.10
Simplify the numerator.
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Step 11.10.1
Multiply by .
Step 11.10.2
Add and .
Step 11.11
Multiply the numerator by the reciprocal of the denominator.
Step 11.12
Multiply .
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Step 11.12.1
Multiply by .
Step 11.12.2
Multiply by .
Step 12
Substitute into the formula and simplify.
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Step 12.1
Apply the product rule to .
Step 12.2
To write as a fraction with a common denominator, multiply by .
Step 12.3
Combine and .
Step 12.4
Combine the numerators over the common denominator.
Step 12.5
Simplify the numerator.
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Step 12.5.1
Move to the left of .
Step 12.5.2
Add and .
Step 12.6
Multiply by .
Step 12.7
To write as a fraction with a common denominator, multiply by .
Step 12.8
Combine and .
Step 12.9
Combine the numerators over the common denominator.
Step 12.10
Simplify the numerator.
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Step 12.10.1
Multiply by .
Step 12.10.2
Add and .
Step 12.11
Multiply the numerator by the reciprocal of the denominator.
Step 12.12
Multiply .
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Step 12.12.1
Multiply by .
Step 12.12.2
Multiply by .
Step 13
Substitute into the formula and simplify.
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Step 13.1
Apply the product rule to .
Step 13.2
To write as a fraction with a common denominator, multiply by .
Step 13.3
Combine and .
Step 13.4
Combine the numerators over the common denominator.
Step 13.5
Simplify the numerator.
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Step 13.5.1
Move to the left of .
Step 13.5.2
Add and .
Step 13.6
Multiply by .
Step 13.7
To write as a fraction with a common denominator, multiply by .
Step 13.8
Combine and .
Step 13.9
Combine the numerators over the common denominator.
Step 13.10
Simplify the numerator.
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Step 13.10.1
Multiply by .
Step 13.10.2
Add and .
Step 13.11
Multiply the numerator by the reciprocal of the denominator.
Step 13.12
Multiply .
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Step 13.12.1
Multiply by .
Step 13.12.2
Multiply by .
Step 14
List the solutions.