Simplify (12- square root of -81)/2
The question is asking for the simplification of a mathematical expression which consists of a subtraction operation, a square root of a negative number, and a division by 2. The square root of a negative number involves the concept of imaginary numbers, since the square root of a negative number cannot be a real number. The task involves simplifying the expression to its simplest form by calculating the square root of -81 and then performing the indicated arithmetic operations (subtraction and division) accordingly.
$\frac{12 - \sqrt{- 81}}{2}$
Answer
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:
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.
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.
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.
Arithmetic Operations: Basic arithmetic operations (addition, subtraction, multiplication, and division) apply to complex numbers similarly to real numbers, keeping in mind the properties of $i$.
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.
