Simplify (1/3+( square root of 7)/6*i)^2

Problem

(1/3+√(,7)/6*i)2

Solution

1. Identify the expression as a binomial square of the form (a+b)2 where a=1/3 and b=√(,7)/6*i

2. Apply the formula (a+b)2=a2+2*a*b+b2 to expand the expression.

a2=(1/3)2=1/9

2*a*b=2⋅1/3⋅√(,7)/6*i=√(,7)/9*i

b2=(√(,7)/6*i)2=7/36*i2

3. Substitute i2=−1 into the third term to simplify the imaginary component.

b2=7/36*(−1)=−7/36

4. Combine the real parts by finding a common denominator for 1/9 and −7/36

1/9=4/36

4/36−7/36=−3/36

5. Simplify the resulting fraction.

−3/36=−1/12

Final Answer

(1/3+√(,7)/6*i)2=−1/12+√(,7)/9*i

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