Problem 32 - College Algebra

Question

In Exercises \(29-44,\) perform the indicated operations and write the result in standard form. $$5 \sqrt{-8}+3 \sqrt{-18}$$

Step-by-Step Solution

Answer

The simplified form of \(5 \sqrt{-8}+3 \sqrt{-18}\) is \(19i\sqrt{2}\).

Step 1: Simplify the Square Roots

The operation inside the square root is multiplication, so it can be split using the property sqrt(a*b) = sqrt(a) * sqrt(b). The square roots of -8 and -18 can be rewritten as: \(5 \sqrt{-1*8} + 3 \sqrt{-1*18}\) = \(5 \sqrt{-1} \sqrt{8} + 3 \sqrt{-1} \sqrt{18}\). The \( \sqrt{-1} \) can be rewritten as \(i\). Therefore: \(5i \sqrt{8} + 3i \sqrt{18}\).

Step 2: Simplify the Radicals

The square roots can be further simplified: \(5i \sqrt{4*2} + 3i \sqrt{9*2}\) = \(5i * 2 \sqrt{2} + 3i * 3 \sqrt{2}\) = \(10i\sqrt{2} + 9i\sqrt{2}\).

Step 3: Combine Like Terms

There are two terms that are alike, they both have \(i\sqrt{2}\). Therefore, add these two together: \(10i\sqrt{2} + 9i\sqrt{2} = 19i\sqrt{2}\).

Key Concepts

Imaginary Numbers

Imaginary numbers may seem intimidating at first, but they're actually quite simple in essence. Imagine you encountered a number under a square root symbol that is negative, like \(-8\). Normally, a square root results in a real number solution, but when negative numbers are under the square root, real numbers don't cut it.
That's when we introduce the imaginary unit, known as \(i\). The imaginary unit \(i\) is defined as the square root of -1. Thus, for any negative number like \(-a\), we use the property \(\sqrt{-a} = i\sqrt{a}\).

• Example: \(\sqrt{-8} = i \sqrt{8}\).

Using imaginary numbers helps in various calculations and extends the traditional number line into the complex plane. Understanding that \(i\) is simply a tool for working with square roots of negative numbers will ease its handling in math operations.

Simplifying Radicals

Simplifying radicals involves reducing a square root expression to its simplest form. Often, this step makes equations and expressions more manageable.
Radicals can be simplified by expressing the number under the square root as a product of two numbers, one of which is a perfect square.
An example is \(\sqrt{8}\), which can be simplified as \(\sqrt{4 \times 2} = 2\sqrt{2}\), since \(4\) is a perfect square.

• The goal is to \identify the largest perfect square factor\ within the number under the radical.
• Similarly, for \(\sqrt{18}\), the largest perfect square is \(9\), allowing us to write it as: \(\sqrt{9 \times 2} = 3\sqrt{2}\).
Solving these carefully makes expressions simpler and helps in further operations like addition or multiplication. By finding a pattern among solutions, you’ll perform tasks with greater speed and ease.

Combining Like Terms

Combining like terms is another essential skill in algebra that simplifies expressions by bringing together terms that are alike. In the context of complex numbers, you need to ensure the imaginary parts and any accompanying radicals match.
Once simplified, such terms can be added or subtracted freely

• Consider terms \(i\sqrt{2}\) occurring repeatedly in an expression.
• For instance, if you have \(10i\sqrt{2} + 9i\sqrt{2}\), both have the same structures (\(i\sqrt{2}\)), making them like terms.

By combining them, you add the coefficients – \(10 + 9 = 19\) – resulting in \(19i\sqrt{2}\). The process is straightforward, but recognizing alike terms is key to maintaining equation accuracy.

Standard Form

The standard form of complex numbers is expressed usually as \(a + bi\), where \(a\) is the real part and \(b\) is the imaginary part involving \(i\).
Even though expressions involving radicals like \(19i\sqrt{2}\) don't directly align, understanding the concept is vital.

• If needed, \(19i\sqrt{2}\) can be perceived as having \(a=0\) and \(b=19\sqrt{2}\), thus fitting into the standard form structure indirectly.
• Expressing in standard form ensures consistency in representation and makes further calculations involving addition, multiplication, or division of complex numbers more straightforward.

Overall, adhering to such standard versions helps maintain uniformity in calculations and communication, especially as complexity in mathematical problems increases.