Question

question 10
simplify to standard form
$3+\sqrt{-81} =$
$\bigcirc$ $3 + 9i$
$\bigcirc$ $3 - 81i$
$\bigcirc$ $3 + 81i$
$\bigcirc$ $3 - 9i$

Explanation:

Step1: Recall the imaginary unit definition

The imaginary unit \( i \) is defined as \( i = \sqrt{-1} \), so we can rewrite \( \sqrt{-81} \) as \( \sqrt{81\times(-1)} \).

Step2: Simplify the square root

Using the property \( \sqrt{ab}=\sqrt{a}\times\sqrt{b} \) (for \( a\geq0, b\geq0 \), and here we extend it for the imaginary case), we have \( \sqrt{81\times(-1)}=\sqrt{81}\times\sqrt{-1} \). Since \( \sqrt{81} = 9 \) and \( \sqrt{-1}=i \), this simplifies to \( 9i \).

Step3: Combine with the real part

Now we substitute back into the original expression \( 3+\sqrt{-81} \), so it becomes \( 3 + 9i \).

Answer:

A. \( 3 + 9i \)