Question

use the imaginary number i to rewrite the expression below as a complex number. simplify all radicals. -√(-27)

Explanation:

Step1: Recall the definition of \(i\)

We know that \(i = \sqrt{-1}\), so we can rewrite \(\sqrt{-27}\) as \(\sqrt{27\times(-1)}\).

Step2: Use the property of square roots

Using the property \(\sqrt{ab}=\sqrt{a}\times\sqrt{b}\) (for \(a\geq0,b\geq0\), here we extend it to complex numbers where \(b = - 1\)), we have \(\sqrt{27\times(-1)}=\sqrt{27}\times\sqrt{-1}\).

Step3: Simplify \(\sqrt{27}\)

We can factor \(27 = 9\times3\), so \(\sqrt{27}=\sqrt{9\times3}=\sqrt{9}\times\sqrt{3}=3\sqrt{3}\).

Step4: Substitute back

Since \(\sqrt{-1}=i\), then \(\sqrt{-27}=3\sqrt{3}i\). And the original expression is \(-\sqrt{-27}\), so \(-\sqrt{-27}=- 3\sqrt{3}i\).

Answer:

\(-3\sqrt{3}i\)