Question

use the imaginary number i to rewrite the expression below as a complex number. simplify all radicals. 12 + \sqrt{-63}

Explanation:

Step1: Rewrite the square root of negative number

Recall that \(\sqrt{-a}=\sqrt{a}\cdot\sqrt{-1}\) for \(a>0\), and \(\sqrt{-1} = i\). So, \(\sqrt{-63}=\sqrt{63}\cdot\sqrt{-1}\).

Step2: Simplify the square root of 63

Factor 63: \(63 = 9\times7\), so \(\sqrt{63}=\sqrt{9\times7}=\sqrt{9}\times\sqrt{7}=3\sqrt{7}\).

Step3: Substitute back and combine

Substitute \(\sqrt{63}=3\sqrt{7}\) and \(\sqrt{-1}=i\) into \(\sqrt{-63}\), we get \(\sqrt{-63}=3\sqrt{7}i\). Then the expression \(12+\sqrt{-63}\) becomes \(12 + 3\sqrt{7}i\).

Answer:

\(12 + 3\sqrt{7}i\)