Simplify square root of 1/3 square root of -27

Problem

√(,1/3√(,−27))

Solution

1. Express the imaginary unit by identifying that the square root of a negative number involves i where i=√(,−1)

√(,−27)=√(,27)⋅i

2. Simplify the inner radical by factoring the radicand into a perfect square and a remainder.

√(,27)=√(,9⋅3)=3√(,3)

3. Substitute the simplified radical back into the original expression.

√(,1/3⋅3√(,3)⋅i)

4. Cancel the common factor of 3 in the numerator and denominator inside the outer radical.

√(,√(,3)⋅i)

5. Apply the property of exponents to rewrite the nested square root, noting that √(,3)=3(1/2) and the outer square root is a power of 1/2

(3(1/2)⋅i)(1/2)

6. Distribute the exponent to both terms inside the parentheses.

3(1/4)⋅i(1/2)

7. Evaluate the square root of i using the identity √(,i)=(1+i)/√(,2)

3(1/4)⋅(1+i)/√(,2)

8. Combine the constants into a single radical expression.

√(4,3)/√(,2)*(1+i)

Final Answer

√(,1/3√(,−27))=√(4,3)/√(,2)*(1+i)

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