Question

Multiply. Write your answer in simplest form.
(squareroot22 - squareroot5) (-squareroot11 + squareroot3)

Answer 1

To multiply the expression \((\sqrt{22} - \sqrt{5})(-\sqrt{11} + \sqrt{3})\), we will use the distributive property (also known as the FOIL method for binomials).

Let's break it down step by step:

\[
(\sqrt{22} - \sqrt{5})(-\sqrt{11} + \sqrt{3}) = \sqrt{22} \cdot (-\sqrt{11}) + \sqrt{22} \cdot \sqrt{3} - \sqrt{5} \cdot (-\sqrt{11}) - \sqrt{5} \cdot \sqrt{3}
\]

Now, calculating each term:

1. \(\sqrt{22} \cdot (-\sqrt{11}) = -\sqrt{22 \cdot 11} = -\sqrt{242}\)
2. \(\sqrt{22} \cdot \sqrt{3} = \sqrt{22 \cdot 3} = \sqrt{66}\)
3. \(-\sqrt{5} \cdot (-\sqrt{11}) = \sqrt{5 \cdot 11} = \sqrt{55}\)
4. \(-\sqrt{5} \cdot \sqrt{3} = -\sqrt{5 \cdot 3} = -\sqrt{15}\)

Now, combine all the terms:

\[
-\sqrt{242} + \sqrt{66} + \sqrt{55} - \sqrt{15}
\]

Next, we simplify \(\sqrt{242}\):

\[
\sqrt{242} = \sqrt{121 \cdot 2} = \sqrt{121} \cdot \sqrt{2} = 11\sqrt{2}
\]

Substituting this back into our expression gives:

\[
-11\sqrt{2} + \sqrt{66} + \sqrt{55} - \sqrt{15}
\]

Thus, the final result in simplest form is:

\[
\boxed{-11\sqrt{2} + \sqrt{66} + \sqrt{55} - \sqrt{15}}
\]