Problem 36 - College Algebra Essentials

Question

In Exercises \(29-44\), perform the indicated operations and write the result in standard form. $$ (-2+\sqrt{-11})^{2} $$

Step-by-Step Solution

Answer

So, the complex number \((-2+\sqrt{-11})^{2}\) in standard form is \(-7 - 4i\sqrt{11}\)

Step 1: Convert the Number Under Root to Complex Number

Replace \(\sqrt{-11}\) with \(i\sqrt{11}\). So, \(-2+ \sqrt{-11}\) becomes \(-2+ i\sqrt{11}\).

Step 2: Apply the Exponent

Now, square \(-2 + i\sqrt{11}\) to get \(( -2 + i\sqrt{11})^2 = (-2)^2 + 2*(-2 * i\sqrt{11}) + (i\sqrt{11})^2\)

Step 3: Simplification

Evaluate and simplify each term to get \(4 -4i\sqrt{11} - 11\)

Step 4: Further Simplification

By combining the real parts, \(4 - 11\) is simplified to get \(-7 -4i\sqrt{11}\)

Key Concepts

Imaginary Unit

The imaginary unit, denoted as \(i\), is a fundamental concept in the world of complex numbers. It is defined as the square root of \(-1\). In mathematical terms, this means \(i^2 = -1\). The introduction of the imaginary unit allows us to extend the real number system to include solutions to equations that have no solutions in the realm of real numbers, such as \(x^2 + 1 = 0\).
Without \(i\), the square root of negative numbers would be undefined in real numbers. Here's a simple explanation of how \(i\) works:

• If \(x^2 = -1\), then \(x = i\).
• Any negative number's square root can be expressed as a multiple of \(i\). For example, \(\sqrt{-11}\) can be rewritten as \(i\sqrt{11}\).

By understanding and using \(i\), mathematicians can express complex numbers, which have both real and imaginary parts, offering a fuller palette to solve various algebraic problems.

Standard Form

The standard form of a complex number is a way of expressing them as \(a + bi\), where \(a\) and \(b\) are real numbers.\(a\) is the real part and \(bi\) is the imaginary part. This form makes complex numbers easy to understand and useful for various calculations.
For example, if we have a complex number like \(-2 + i\sqrt{11}\), it is expressed in standard form with the real part being \(-2\) and the imaginary part being \(i\sqrt{11}\).

• It's simpler to perform operations like addition or subtraction between complex numbers in this form.
• In algebra, expressing numbers this way allows us to clearly identify and differentiate between the real and imaginary components.

Understanding the standard form is essential for manipulating complex numbers and for applications ranging from simple algebra to more advanced calculus.

Algebraic Operations

Algebraic operations with complex numbers follow some rules that are similar to those for real numbers but also include considerations for the imaginary unit \(i\). Here's how they work:

• Addition and Subtraction: Add or subtract the real parts and the imaginary parts separately. For example, \((a + bi) + (c + di) = (a+c) + (b+d)i\).
• Multiplication: Use the distributive property to multiply complex numbers. Remember, \(i^2 = -1\). For example, \((a + bi)(c + di) = ac + adi + bci + bdi^2\), which simplifies to \((ac-bd) + (ad+bc)i\).
• Squaring: When squaring a complex number like \(-2 + i\sqrt{11}\), use the formula: \((a + bi)^2 = a^2 + 2abi + (bi)^2\)

Performing these operations allows us to solve complex number equations and express the result in standard form, giving a clearer picture of the number's structure.

Complex Conjugate

The complex conjugate of a complex number is obtained by changing the sign of its imaginary part. If we have a complex number \(a + bi\), its complex conjugate is \(a - bi\).
Complex conjugates have some interesting properties that make them especially useful in mathematics:

• Multiplying a complex number by its conjugate always results in a real number: \((a + bi)(a - bi) = a^2 + b^2\).
• They can be used to simplify division of complex numbers: To divide by a complex number, multiply the numerator and the denominator by the conjugate of the denominator.
• Conjugates preserve the magnitude: While they change the sign of the imaginary part, the overall size or magnitude of the complex number remains unchanged.

Understanding and utilizing complex conjugates allow for efficient handling of complex equations and simplification of their expressions.