Question
Simplify by using the imaginary unit \( i \) \( \sqrt{-8} \cdot \sqrt{-2} \) \( \sqrt{-8} \cdot \sqrt{-2}=\square \) (Simplify your answer. Express complex numbers in terms of \( i \).)
Ask by Munoz Colon. in the United States
Jan 27,2025
Real Tutor Solution
Tutor-Verified Answer
Answer
\( \sqrt{-8} \cdot \sqrt{-2} = -4 \)
Solution
Calculate or simplify the expression \( \sqrt{-8} \cdot \sqrt{-2} \).
Calculate the value by following steps:
- step0: Calculate:
\(\sqrt{-8}\times \sqrt{-2}\)
- step1: Simplify the root:
\(2\sqrt{2}\times i\sqrt{-2}\)
- step2: Simplify the root:
\(2\sqrt{2}\times i\sqrt{2}\times i\)
- step3: Multiply:
\(2\sqrt{2}\times \sqrt{2}\times i^{2}\)
- step4: Multiply:
\(4i^{2}\)
- step5: Use \(i^2=-1\) to transform the expression\(:\)
\(4\left(-1\right)\)
- step6: Calculate:
\(-4\)
The simplified form of \( \sqrt{-8} \cdot \sqrt{-2} \) is \( -4 \).
Reviewed and approved by the UpStudy tutoring team
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Simplify this solution Mind Expander
To simplify \( \sqrt{-8} \cdot \sqrt{-2} \), we start by rewriting each square root in terms of \( i \). First, we have: \[ \sqrt{-8} = \sqrt{8} \cdot \sqrt{-1} = \sqrt{8} \cdot i = 2\sqrt{2} \cdot i \] Next, we simplify \( \sqrt{-2} \): \[ \sqrt{-2} = \sqrt{2} \cdot \sqrt{-1} = \sqrt{2} \cdot i \] Now, we can multiply the two results: \[ \sqrt{-8} \cdot \sqrt{-2} = (2\sqrt{2} \cdot i) \cdot (\sqrt{2} \cdot i) = 2\sqrt{2} \cdot \sqrt{2} \cdot i^2 \] Since \( i^2 = -1 \), we substitute that in: \[ 2\sqrt{2} \cdot \sqrt{2} \cdot i^2 = 2 \cdot 2 \cdot (-1) = 4 \cdot (-1) = -4 \] Thus, the simplified result is: \[ \sqrt{-8} \cdot \sqrt{-2} = -4 \]
