Question

use the imaginary number i to rewrite the expression below as a complex number. simplify all radicals. 11 + \sqrt{-56}

Explanation:

Step1: Simplify the square root of -56

We know that \(\sqrt{-a}=\sqrt{a}\times\sqrt{-1}=i\sqrt{a}\) for \(a>0\). So, \(\sqrt{-56}=i\sqrt{56}\).

Step2: Simplify \(\sqrt{56}\)

Factor 56: \(56 = 4\times14\), and \(\sqrt{4\times14}=\sqrt{4}\times\sqrt{14}=2\sqrt{14}\).

Step3: Substitute back

So, \(\sqrt{-56}=i\times2\sqrt{14}=2i\sqrt{14}\).

Step4: Combine with the real part

The original expression is \(11+\sqrt{-56}\), substituting the simplified radical, we get \(11 + 2i\sqrt{14}\).

Answer:

\(11 + 2i\sqrt{14}\)