Problem 49 - College Algebra

Question

Perform the indicated operation(s) 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: Simplify the Square Roots

Simplify each square root as square root of a positive number times \(i\). \(\sqrt{-16}\) can be rewritten as \(\sqrt{16} \cdot \sqrt{-1}\), which is \(4i\). Similarly, \(\sqrt{-81}\) becomes \(9i\). The expression becomes \(5(4i) + 3(9i)\).

Step 2: Multiply the Real Numbers by \(i\)

Multiply the real coefficients by \(i\). This gives \(20i + 27i\).

Step 3: Combine Like Terms

Combine the terms with \(i\) to get the simplified form of the expression. The result is \(47i\).

Key Concepts

Imaginary Numbers

When dealing with imaginary numbers, it's important to understand the basic concept behind them. An imaginary number is formed when we take the square root of a negative number. The imaginary unit is denoted by the letter \(i\), where \(i\) is defined as \(\sqrt{-1}\). This means whenever you encounter the square root of a negative number, you can convert it into a real number multiplied by \(i\).
For example:

• \(\sqrt{-1} = i\)
• \(\sqrt{-4} = 2i\) because \(\sqrt{4} = 2\)
• \(\sqrt{-9} = 3i\) because \(\sqrt{9} = 3\)

Imaginary numbers are an extension of the real number system. They help us solve problems involving square roots of negative numbers, where no real solutions exist. This becomes especially useful in fields like engineering and physics.

Simplifying Square Roots

Simplifying square roots, especially those of negative numbers, is an essential skill in math. The process essentially allows us to break down more complex expressions into simpler, more manageable terms.
First, identify whether the square root involves a negative number. If it does, rewrite the expression using \(i\). For example, \(\sqrt{-16}\) can be decomposed into \(\sqrt{16} \cdot \sqrt{-1}\). This simplifies to \(4i\) because \(\sqrt{16}=4\) and \(\sqrt{-1}=i\).
The same applies to \(\sqrt{-81}\), which breaks down into \(\sqrt{81} \cdot \sqrt{-1}\), which equals \(9i\) as \(\sqrt{81}=9\).
The procedure of simplifying these allows us to express complex operations in terms of real numbers and imaginary units, hence enabling us to move forward with calculations methodically.

Standard Form of Complex Numbers

The standard form of complex numbers is a way to write numbers that have both real and imaginary parts. This form looks like \(a + bi\), where \(a\) is the real part and \(b\) is the imaginary part. The concept is straightforward:

• If you have a complex number such as \(47i\), it means the real part \(a\) is zero, and the imaginary part \(b\) is 47, giving us \(0 + 47i\).
• In simpler operations, this standard form facilitates the combination of like terms or the performance of arithmetic operations, like addition and subtraction.

Using the standard form makes it easier to understand and visualize complex numbers, especially when dealing with equations or systems involving both real and imaginary components.