Question

use the imaginary number i to rewrite the expression below as a complex number. simplify all radicals. 2 + \sqrt{-45}

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, $\sqrt{-45}=\sqrt{45}\cdot\sqrt{-1}$.

Step2: Simplify the square root of 45

Factor 45: $45 = 9\times5$, so $\sqrt{45}=\sqrt{9\times5}=\sqrt{9}\times\sqrt{5}=3\sqrt{5}$.

Step3: Substitute back and combine

Since $\sqrt{-45}=\sqrt{45}\cdot i = 3\sqrt{5}i$, then the original expression $2+\sqrt{-45}$ becomes $2 + 3\sqrt{5}i$.

Answer:

$2 + 3\sqrt{5}i$