Simplify (12- square root of -81)/2

Solution:

Step 1: Extract the imaginary unit $i$.

Step 1.1

Express $-81$ as $-1\times 81$. Thus, the expression becomes $\frac{12-\sqrt{-1\times 81}}{2}$.

Step 1.2

Decompose $\sqrt{-1\times 81}$ into $\sqrt{-1}\cdot \sqrt{81}$. The expression now reads $\frac{12-(\sqrt{-1}\cdot \sqrt{81})}{2}$.

Step 1.3

Substitute $\sqrt{-1}$ with $i$. The expression simplifies to $\frac{12-(i\cdot \sqrt{81})}{2}$.

Step 2: Simplify the numerator.

Step 2.1

Rewrite $81$ as $9^2$. The expression is now $\frac{12-i\cdot \sqrt{9^2}}{2}$.

Step 2.2

Extract terms from under the radical, assuming they represent positive real numbers. This gives us $\frac{12-i\cdot 9}{2}$.

Step 2.3

Combine the terms in the numerator. The final simplified form is $\frac{12-9i}{2}$.

Knowledge Notes:

To simplify the given expression, we follow a systematic approach that involves understanding complex numbers and their properties. Here are the relevant knowledge points:

1. Imaginary Unit ($i$): The imaginary unit $i$ is defined as the square root of -1. In complex numbers, $i$ is used to express the square root of negative numbers.

2. Complex Numbers: A complex number is a number that can be expressed in the form $a+bi$, where $a$ and $b$ are real numbers, and $i$ is the imaginary unit.

3. Simplifying Square Roots: When simplifying the square root of a negative number, we extract $i$ to handle the negative part and then simplify the square root of the positive component as we would with any real number.

4. Arithmetic Operations: Basic arithmetic operations (addition, subtraction, multiplication, and division) apply to complex numbers similarly to real numbers, keeping in mind the properties of $i$.

5. Radicals: When dealing with radicals, such as $\sqrt{a^2}$, where $a$ is a real number, the result is the absolute value of $a$. This is because we assume the principal square root, which is always non-negative.

By applying these concepts, we can simplify the given expression involving both real numbers and the imaginary unit.