Question

For Exercises \(1-74,\) simplify. $$\\{2[14-11]+\sqrt{(4)(-9)}\\}-[12-6 \cdot 9]$$

Short answer

Question: Simplify the given expression: $$\{2[14-11]+\sqrt{(4)(-9)}\}-[12-6 \cdot 9]$$ Answer: The simplified expression is $$\: 48+6i$$.

Step-by-step solution

Inside the brackets and square root

Calculate the values inside the brackets and square root first. $$\{2[14-11]+\sqrt{(4)(-9)}\}-[12-6 \cdot 9] = \{2[3]+\sqrt{(-36)}\}-[12-54]$$

Evaluate the square root

Next, evaluate the square root of -36. Since the square root of a negative number is a complex number, we express the result as a multiple of the imaginary unit \(i\) (where \(i^2 = -1\)). $$\sqrt{-36}=6i$$ The expression becomes: $$\{2[3]+6i\}-[12-54]$$

Perform the multiplication

Multiply the numbers outside the brackets. $$\{6+6i\}-[12-54]$$

Inside the brackets

Calculate the value inside the remaining bracket. $$\{6+6i\}-[-42]$$

Perform subtraction

Finally, perform subtraction between the numbers. $$6+6i+42=48+6i$$ So, the simplified expression is: $$\: \boldsymbol{48+6i}$$

Key concepts

Complex Numbers

Complex numbers are a fundamental concept in mathematics that combines real numbers with imaginary numbers. They are expressed in the form \(a + bi\), where \(a\) is the real part and \(b\) is the imaginary part.

Some key properties of complex numbers include:

• Every complex number is made up of a real part and an imaginary part.
• The imaginary unit, denoted as \(i\), is defined by the property \(i^2 = -1\).
• Complex numbers can be added, subtracted, multiplied, and divided similar to real numbers, with additional rules for the imaginary parts.

The usage of complex numbers extends to many areas such as engineering, physics, and applied mathematics, where they help to handle problems involving two-dimensional spaces.

Simplifying Expressions

Simplifying expressions is a key part of mathematics that involves reducing an expression to its most compact form. This makes calculations easier and more understandable.

Here are the main steps for simplifying expressions:

• First, work within parentheses or brackets to evaluate any operations inside them.
• Execute operations involving exponents or roots, like finding square roots.
• Next, carry out any multiplications or divisions in the expression.
• Finally, perform additions or subtractions, simplifying fully to a numerical result or a simplified algebraic expression.

Simplifying helps in making expressions easier to visualize and solve, especially when dealing with complex numbers and calculations.

Imaginary Unit

The imaginary unit \(i\) is a mathematical constant that is crucial for dealing with square roots of negative numbers. It is defined by the equation \(i^2 = -1\).

This unique property sets up some interesting applications and rules:

• It allows mathematicians and engineers to move beyond the limitations of real numbers, enabling the square rooting of negative numbers.
• Any real number can multiply with \(i\) to form purely imaginary numbers, such as \(4i\), \(-3i\), or \(bi\).
• The inclusion of the imaginary part with real numbers creates complex numbers, vital in solving quadratic equations with no real roots.

The imaginary unit has many applications in fields like quantum physics and electrical engineering, where complex numbers help model and solve real-world problems.