Linear Algebra Examples

Find the Fourth Roots of a Complex Number 3(cos(pi)+isin(pi))
Step 1
Calculate the distance from to the origin using the formula .
Step 2
Simplify .
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Step 2.1
Apply the reference angle by finding the angle with equivalent trig values in the first quadrant. Make the expression negative because cosine is negative in the second quadrant.
Step 2.2
The exact value of is .
Step 2.3
Multiply .
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Step 2.3.1
Multiply by .
Step 2.3.2
Multiply by .
Step 2.4
Raise to the power of .
Step 2.5
Apply the reference angle by finding the angle with equivalent trig values in the first quadrant.
Step 2.6
The exact value of is .
Step 2.7
Multiply by .
Step 2.8
Raising to any positive power yields .
Step 2.9
Add and .
Step 2.10
Rewrite as .
Step 2.11
Pull terms out from under the radical, assuming positive real numbers.
Step 3
Calculate reference angle .
Step 4
Simplify .
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Step 4.1
Cancel the common factor of .
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Step 4.1.1
Cancel the common factor.
Step 4.1.2
Rewrite the expression.
Step 4.2
Simplify the numerator.
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Step 4.2.1
Apply the reference angle by finding the angle with equivalent trig values in the first quadrant.
Step 4.2.2
The exact value of is .
Step 4.3
Simplify the denominator.
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Step 4.3.1
Apply the reference angle by finding the angle with equivalent trig values in the first quadrant. Make the expression negative because cosine is negative in the second quadrant.
Step 4.3.2
The exact value of is .
Step 4.3.3
Multiply by .
Step 4.4
Simplify the expression.
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Step 4.4.1
Move the negative one from the denominator of .
Step 4.4.2
Multiply by .
Step 4.5
The absolute value is the distance between a number and zero. The distance between and is .
Step 4.6
The exact value of is .
Step 5
Find the quadrant.
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Step 5.1
Apply the reference angle by finding the angle with equivalent trig values in the first quadrant. Make the expression negative because cosine is negative in the second quadrant.
Step 5.2
The exact value of is .
Step 5.3
Multiply .
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Step 5.3.1
Multiply by .
Step 5.3.2
Multiply by .
Step 5.4
Apply the reference angle by finding the angle with equivalent trig values in the first quadrant.
Step 5.5
The exact value of is .
Step 5.6
Multiply by .
Step 5.7
Since the x-coordinate is negative and the y-coordinate is , the point is located on x-axis between the second and third quadrants. The quadrants are labeled in counter-clockwise order, starting in the upper-right.
Between Quadrant and
Between Quadrant and
Step 6
Use the formula to find the roots of the complex number.
,
Step 7
Substitute , , and into the formula.
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Step 7.1
Combine and .
Step 7.2
Combine and .
Step 7.3
Combine and .
Step 7.4
Combine and .
Step 7.5
Remove parentheses.
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Step 7.5.1
Remove parentheses.
Step 7.5.2
Remove parentheses.
Step 7.5.3
Remove parentheses.
Step 7.5.4
Remove parentheses.
Step 7.5.5
Remove parentheses.
Step 7.5.6
Remove parentheses.
Step 7.5.7
Remove parentheses.
Step 8
Substitute into the formula and simplify.
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Step 8.1
Remove parentheses.
Step 8.2
Multiply .
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Step 8.2.1
Multiply by .
Step 8.2.2
Multiply by .
Step 9
Substitute into the formula and simplify.
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Step 9.1
Remove parentheses.
Step 9.2
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 11
Substitute into the formula and simplify.
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Step 11.1
Remove parentheses.
Step 11.2
Multiply by .
Step 12
List the solutions.