Evaluate

Problem

(7−√(,−49))/(7+√(,−49))

Solution

1. Express the square roots of negative numbers in terms of the imaginary unit i where √(,−49)=√(,49)⋅√(,−1)=7*i

(7−7*i)/(7+7*i)

2. Simplify the fraction by factoring out the greatest common factor, which is 7 from both the numerator and the denominator.

(7*(1−i))/(7*(1+i))

3. Cancel the common factor of 7 to reduce the expression.

(1−i)/(1+i)

4. Rationalize the denominator by multiplying both the numerator and the denominator by the complex conjugate of the denominator, which is 1 - i$.

((1−i)*(1−i))/((1+i)*(1−i))

5. Expand the numerator using the FOIL method and the denominator using the difference of squares pattern.

(1−i−i+i2)/(1−i2)

6. Substitute i2=−1 into the expression and combine like terms.

(1−2*i−1)/(1−(−1))

7. Simplify the resulting fraction.

(−2*i)/2

8. Divide to find the final simplified value.

−i

Final Answer

(7−√(,−49))/(7+√(,−49))=−i

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