Question

use the imaginary number i to rewrite the expression below as a complex number. simplify all radicals.
$-sqrt{-72}$

Explanation:

Step1: Recall the definition of imaginary unit

We know that \( i = \sqrt{-1} \), so we can rewrite \( \sqrt{-72} \) as \( \sqrt{72\times(-1)} \).

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), we have \( \sqrt{72\times(-1)}=\sqrt{72}\times\sqrt{-1} \).

Step3: Simplify \( \sqrt{72} \)

We can factor 72 as \( 36\times2 \), so \( \sqrt{72}=\sqrt{36\times2}=\sqrt{36}\times\sqrt{2}=6\sqrt{2} \).

Step4: Substitute back and consider the negative sign

Since \( \sqrt{-1} = i \), then \( \sqrt{-72}=6\sqrt{2}i \). Now, the original expression is \( -\sqrt{-72} \), so we have \( - 6\sqrt{2}i \).

Answer:

\( -6\sqrt{2}i \)