Questions: Simplify the following radical. [ sqrt-300 ]

Solution Steps

To simplify the square root of a negative number, we need to express it in terms of imaginary numbers. The square root of \(-1\) is represented by the imaginary unit \(i\). Therefore, \(\sqrt{-300}\) can be rewritten as \(i \times \sqrt{300}\). Next, simplify \(\sqrt{300}\) by finding its prime factors and simplifying the radical.

We start with the expression \( \sqrt{-300} \). Since the square root of a negative number involves the imaginary unit \( i \), we can rewrite this as: \[ \sqrt{-300} = i \cdot \sqrt{300} \]

Next, we simplify \( \sqrt{300} \). The prime factorization of \( 300 \) is: \[ 300 = 2^2 \cdot 3^1 \cdot 5^2 \] Using this factorization, we can simplify \( \sqrt{300} \): \[ \sqrt{300} = \sqrt{2^2 \cdot 3 \cdot 5^2} = \sqrt{2^2} \cdot \sqrt{3} \cdot \sqrt{5^2} = 2 \cdot 5 \cdot \sqrt{3} = 10\sqrt{3} \]

Now, substituting back into our expression, we have: \[ \sqrt{-300} = i \cdot 10\sqrt{3} \]

Final Answer