Linear Algebra Examples

Find the Cube Roots of a Complex Number 1+i
Step 1
Calculate the distance from to the origin using the formula .
Step 2
Simplify .
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Step 2.1
One to any power is one.
Step 2.2
One to any power is one.
Step 2.3
Add and .
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 first quadrant because and are both positive. The quadrants are labeled in counter-clockwise order, starting in the upper-right.
Quadrant
Step 6
is in the first quadrant.
Step 7
Use the formula to find the roots of the complex number.
,
Step 8
Substitute , , and into the formula.
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Step 8.1
Combine and .
Step 8.2
Combine and .
Step 8.3
Combine and .
Step 8.4
Combine and .
Step 8.5
Remove parentheses.
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Step 8.5.1
Remove parentheses.
Step 8.5.2
Remove parentheses.
Step 8.5.3
Remove parentheses.
Step 8.5.4
Remove parentheses.
Step 8.5.5
Remove parentheses.
Step 8.5.6
Remove parentheses.
Step 8.5.7
Remove parentheses.
Step 8.5.8
Remove parentheses.
Step 9
Substitute into the formula and simplify.
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Step 9.1
Remove parentheses.
Step 9.2
Multiply .
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Step 9.2.1
Multiply by .
Step 9.2.2
Multiply by .
Step 9.3
Add and .
Step 9.4
Multiply the numerator by the reciprocal of the denominator.
Step 9.5
Multiply .
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Step 9.5.1
Multiply by .
Step 9.5.2
Multiply by .
Step 10
Substitute into the formula and simplify.
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Step 10.1
Remove parentheses.
Step 10.2
Multiply by .
Step 10.3
To write as a fraction with a common denominator, multiply by .
Step 10.4
Combine and .
Step 10.5
Combine the numerators over the common denominator.
Step 10.6
Simplify the numerator.
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Step 10.6.1
Multiply by .
Step 10.6.2
Add and .
Step 10.7
Multiply the numerator by the reciprocal of the denominator.
Step 10.8
Cancel the common factor of .
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Step 10.8.1
Factor out of .
Step 10.8.2
Cancel the common factor.
Step 10.8.3
Rewrite the expression.
Step 11
Substitute into the formula and simplify.
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Step 11.1
Remove parentheses.
Step 11.2
Multiply by .
Step 11.3
To write as a fraction with a common denominator, multiply by .
Step 11.4
Combine and .
Step 11.5
Combine the numerators over the common denominator.
Step 11.6
Simplify the numerator.
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Step 11.6.1
Multiply by .
Step 11.6.2
Add and .
Step 11.7
Multiply the numerator by the reciprocal of the denominator.
Step 11.8
Multiply .
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Step 11.8.1
Multiply by .
Step 11.8.2
Multiply by .
Step 12
List the solutions.