Evaluate square root of -14 square root of -2

Problem

√(,−14)√(,−2)

Solution

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

2. Rewrite the first term using the property √(,−a)=i√(,a) for a>0

√(,−14)=i√(,14)

3. Rewrite the second term using the same property.

√(,−2)=i√(,2)

4. Multiply the two expressions together.

i√(,14)⋅i√(,2)=i2√(,14⋅2)

5. Substitute i2=−1 and simplify the radicand.

−1⋅√(,28)

6. Simplify the radical by factoring out the largest perfect square.

−1⋅√(,4⋅7)

7. Extract the square root of the perfect square.

−1⋅2√(,7)

Final Answer

√(,−14)√(,−2)=−2√(,7)

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