Question

what is the imaginary part of the simplest form of the complex number 9.2 - 3√(-8)? (1 poi

Explanation:

Step1: Simplify the square root of negative number

Recall that \( \sqrt{-a}=\sqrt{a}\cdot\sqrt{-1}=i\sqrt{a} \) for \( a>0 \). So for \( \sqrt{-8} \), we have \( \sqrt{-8}=\sqrt{8}\cdot i = 2\sqrt{2}i \) (since \( \sqrt{8}=\sqrt{4\times2}=2\sqrt{2} \)).

Step2: Substitute back into the complex number

The complex number is \( 9.2 - 3\sqrt{-8} \). Substituting \( \sqrt{-8}=2\sqrt{2}i \) into it, we get \( 9.2-3\times(2\sqrt{2}i)=9.2 - 6\sqrt{2}i \).

Step3: Identify the imaginary part

In the standard form of a complex number \( a + bi \), where \( a \) is the real part and \( b \) is the imaginary part (the coefficient of \( i \)). Here, the complex number is \( 9.2-6\sqrt{2}i \), so the imaginary part is \( - 6\sqrt{2} \) (or we can write it as \( -6\sqrt{2} \), and if we calculate the numerical value, \( \sqrt{2}\approx1.414 \), so \( -6\times1.414\approx - 8.484 \), but the exact form is \( -6\sqrt{2} \)).

Answer:

\( -6\sqrt{2} \) (or approximately \( -8.48 \))