Problem 31 - College Algebra

Question

In Exercises \(29-44,\) perform the indicated operations and write the result in standard form. $$5 \sqrt{-16}+3 \sqrt{-81}$$

Step-by-Step Solution

Answer

The result in standard form is \(47i\).

Step 1: Understanding the representation of complex numbers

Complex numbers are often represented in the form \(a+bi\), where \(a\) and \(b\) are real numbers, and \(i\) is the imaginary unit with the property that \(i^2 = -1\). Real numbers are the values that can represent a quantity along a continuous line, while the \(i=\sqrt{-1}\) is the imaginary unit and b is an imaginary part of the complex number. Recognize that \(\sqrt{-16}\) and \(\sqrt{-81}\) signify the presence of the imaginary unit \(i\).

Step 2: Break down the square roots of negatives

Firstly, break down each square root separately. \(5 \sqrt{-16}\) can be written as \$5 *(\sqrt{16} * \sqrt{-1})\$. Since \(\sqrt{16}\) is 4 and \(\sqrt{-1}\) is \(i\), \(5 \sqrt{-16}\) can be expressed as \(5*4*i\), which simplifies to \(20i\). Similarly, \(3 \sqrt{-81}\) can be written as \(3* (\sqrt{81} * \sqrt{-1})\). Since \(\sqrt{81}\) is 9 and \(\sqrt{-1}\) is \(i\), \(3 \sqrt{-81}\) can be expressed as \(3*9*i\), which simplifies to \(27i\).

Step 3: Perform the addition

Now, perform the addition \(20i + 27i\). Adding like terms gives \(47i\).

Key Concepts

Imaginary Unit

Understanding the imaginary unit is a fundamental step in dealing with complex numbers. It is represented by the symbol 'i' and is defined by its distinctive property: \(i^2 = -1\). The term 'imaginary' might suggest that these numbers don't exist, but this is far from the truth. They are a crucial part of mathematics and have real-world applications in fields like electrical engineering and quantum physics.

The imaginary unit allows us to take the square root of negative numbers, which is not possible with real numbers alone. For example, \(\sqrt{-1}\) is represented as 'i', \(\sqrt{-4}\) as '2i', and so on. Consequently, any real number multiplied by 'i' gives us an imaginary number.

Standard Form

The standard form for writing complex numbers is \(a + bi\), where 'a' and 'b' are real numbers, and 'bi' is the imaginary part. The beauty of this form is its clarity in representing both the real and imaginary components of complex numbers.

Let's take the expression from the exercise, \(5 \sqrt{-16} + 3 \sqrt{-81}\), and apply the standard form. Each square root of a negative number includes an imaginary unit 'i', resulting in real coefficients multiplied by 'i'. After simplifying, the terms become \(20i\) and \(27i\), and in standard form, their sum is presented as \(47i\), clearly showing the imaginary part of the complex number.

Operations with Complex Numbers

Performing operations with complex numbers, such as addition, subtraction, multiplication, and division, follows specific rules to ensure we stay within the realm of complex numbers.

For addition and subtraction, as shown in the original exercise, you combine like terms. Imaginary units combine with other imaginary units and real numbers combine with other real numbers. Multiplication and division are more intricate, often involving the use of the FOIL method or conjugates to resolve the products and quotients.

Simplifying Imaginary Numbers

When multiplying complex numbers and simplifying expressions with imaginary units, it's important to remember the defining property of the imaginary unit, \(i^2 = -1\). This identity helps in reducing powers of 'i' and allows simplification of the result into standard form.

Example

If we multiply \( (3+4i)(2+i) \), we apply the distributive property: \( 3\cdot 2 + 3\cdot i + 4i\cdot 2 + 4i\cdot i = 6 + 3i + 8i + 4i^2 = 6 + 11i + 4(-1) \), resulting in the standard form \( 2 + 11i \). Combining our knowledge of 'i' and these operations, complex numbers become much more manageable and less intimidating!