To simplify the expression
[tex]\[
-3 i \sqrt{-40}
\][/tex]
we can follow these steps:
1. Simplify the square root of [tex]\(-40\)[/tex]:
We know that the square root of a negative number can be expressed in terms of the imaginary unit [tex]\( i \)[/tex]. Recall that:
[tex]\[
\sqrt{-1} = i
\][/tex]
Therefore:
[tex]\[
\sqrt{-40} = \sqrt{40} \cdot \sqrt{-1} = \sqrt{40} \cdot i
\][/tex]
2. Simplify [tex]\(\sqrt{40}\)[/tex]:
[tex]\[
\sqrt{40} \approx 6.324555320336759
\][/tex]
Hence:
[tex]\[
\sqrt{-40} = 6.324555320336759 \cdot i
\][/tex]
3. Insert this back into the original expression:
[tex]\[
-3 i \sqrt{-40} = -3 i \cdot (6.324555320336759 \cdot i)
\][/tex]
4. Multiply the terms:
Recall that [tex]\( i^2 = -1 \)[/tex], so:
[tex]\[
-3 i \cdot 6.324555320336759 \cdot i = -3 \cdot 6.324555320336759 \cdot i^2
\][/tex]
5. Simplify using [tex]\( i^2 = -1 \)[/tex]:
[tex]\[
-3 \cdot 6.324555320336759 \cdot (-1) = 3 \cdot 6.324555320336759
\][/tex]
6. Calculate the final result:
[tex]\[
3 \cdot 6.324555320336759 = 18.973665961010275 \approx 18.97
\][/tex]
The simplified form of the expression [tex]\(-3 i \sqrt{-40}\)[/tex] is:
[tex]\[
18.97
\][/tex]
