Question

question
simplify the expression to ( a + bi ) form:
( sqrt{100} + sqrt{-40} + sqrt{25} - sqrt{-10} )

Explanation:

Step1: Simplify real square roots

Simplify \(\sqrt{100}\) and \(\sqrt{25}\). We know that \(\sqrt{100} = 10\) and \(\sqrt{25}=5\). So the expression becomes \(10 + \sqrt{-40}+5 - \sqrt{-10}\).

Step2: Simplify imaginary square roots

Recall that \(\sqrt{-x}=\sqrt{x}\cdot i\) for \(x>0\). So, \(\sqrt{-40}=\sqrt{40}\cdot i = \sqrt{4\times10}\cdot i=2\sqrt{10}\cdot i\) and \(\sqrt{-10}=\sqrt{10}\cdot i\). Substituting these into the expression, we get \(10 + 2\sqrt{10}i+5-\sqrt{10}i\).

Step3: Combine like terms

Combine the real parts (\(10 + 5\)) and the imaginary parts (\(2\sqrt{10}i-\sqrt{10}i\)). The real part sum is \(15\) and the imaginary part sum is \((2\sqrt{10}-\sqrt{10})i=\sqrt{10}i\). So the simplified form is \(15+\sqrt{10}i\).

Answer:

\(15 + \sqrt{10}i\)