Question

practice assignment 2.4 complex numbers
score: 140/250 answered: 14/25
question 15
evaluate the expression \\(\sqrt{-8}\sqrt{-72}\\) and write the result in the form \\(a + bi\\).
the real number \\(a\\) equals
the real number \\(b\\) equals
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Explanation:

Step1: Express square roots of negative numbers

Recall that \(\sqrt{-x}=i\sqrt{x}\) for \(x>0\). So, \(\sqrt{-8}=i\sqrt{8}\) and \(\sqrt{-72}=i\sqrt{72}\).
Then \(\sqrt{-8}\sqrt{-72}=(i\sqrt{8})(i\sqrt{72})\).

Step2: Use the property \(i^2 = - 1\) and multiply the radicals

By the property \(i^2=-1\) and \(\sqrt{a}\cdot\sqrt{b}=\sqrt{ab}\) (\(a,b\geq0\)), we have \((i\sqrt{8})(i\sqrt{72})=i^{2}\sqrt{8\times72}\).
Since \(i^{2}=-1\) and \(8\times72 = 576\), then \(i^{2}\sqrt{8\times72}=-\sqrt{576}\).

Step3: Simplify the square - root

We know that \(\sqrt{576} = 24\), so \(-\sqrt{576}=-24+0i\).

Answer:

The real number \(a=-24\), the real number \(b = 0\).