Problem 50 - College Algebra

Question

Perform the indicated operation(s) and write the result in standard form. $$ 5 \sqrt{-8}+3 \sqrt{-18} $$

Step-by-Step Solution

Answer

The solution to the problem \( 5 \sqrt{-8}+3 \sqrt{-18} \) is \( 19i \sqrt{2} \).

Step 1: Identify the square root of negative numbers

The square root of a negative number is represented in terms of \( i \), where \( i^2 = -1 \). Thus, we can rewrite \( 5 \sqrt{-8} + 3 \sqrt{-18} \) as \( 5i \sqrt{8} + 3i \sqrt{18} \)

Step 2: Simplify the radical expressions

The next step is to simplify the radical expressions under the square root. Yielding \( 5i \sqrt{4*2} + 3i \sqrt{9*2} = 5i \sqrt{4} \sqrt{2} + 3i \sqrt{9} \sqrt{2} = 5i * 2 \sqrt{2} + 3i * 3 \sqrt{2} = 10i \sqrt{2} + 9i \sqrt{2} \)

Step 3: Perform the indicated operation - addition

Now, we just simply add \( 10i \sqrt{2} \) and \( 9i \sqrt{2} \). Which equals \( (10i + 9i) \sqrt{2} = 19i \sqrt{2} \)

Key Concepts

Complex Numbers Algebra

Understanding complex numbers is foundational in exploring higher mathematics. Complex numbers are composed of a real part and an imaginary part, expressed as a + bi, where a is the real part, b is the coefficient of the imaginary part, and i represents the square root of -1.

When performing algebraic operations with complex numbers, one treats the real and imaginary parts separately. Addition or subtraction involves combining real parts with real parts and imaginary parts with imaginary parts. Multiplication and division, however, require the use of the distributive property and sometimes conjugates to simplify expressions.

In the exercise 5 \( \sqrt{-8} \) + 3 \( \sqrt{-18} \), we first recognize the presence of negative radicands, leading us to convert them into complex numbers by factoring out the imaginary unit i. We then simplify the roots and finally combine like terms to arrive at a simplified expression in standard form.

Radicals and Complex Numbers

Combining radicals with complex numbers often confuses students, but it follows a logical process. Radicals, or roots, are the opposite of raising a number to a power. When working with negative numbers under a radical, we invoke the concept of imaginary numbers.

To simplify radicals containing negative numbers within complex expressions, factor out the negative as an imaginary unit, i. For example, \( \sqrt{-8} \) becomes \( i \sqrt{8} \), where \( i \) is the imaginary unit and \( \sqrt{8} \) can be further simplified.

Once the radicals are simplified to their prime factors, such as \( \sqrt{4} \cdot \sqrt{2} \), we use properties of square roots to break them down into simpler forms. In practice, this means converting them into more straightforward expressions, which allows for easy addition or subtraction with other similar terms, as seen in the exercise provided.

Imaginary Numbers i

The concept of the imaginary unit i is a stepping stone to a broader understanding of complex numbers. Imaginary numbers arise when we look for square roots of negative numbers, which do not exist in the set of real numbers.

By definition, \( i \) is the square root of -1. This definition allows us to operate with negative radicands by representing them as multiples of i. When squared, \( i \) returns -1, i.e., \( i^2 = -1 \).

In algebraic operations, the power of \( i \) follows a pattern: \( i^0 = 1, i^1 = i, i^2 = -1, i^3 = -i, i^4 = 1 \), and so on, repeating every four terms. This cyclical behavior is key in simplifying expressions where higher powers of i are present. As illustrated in the exercise, understanding and managing the imaginary unit i is crucial when working with complex numbers and simplifying radical expressions containing negative numbers.