Problem 28 - Algebra for College Students

Question

Write each of the following in terms of \(i\) and simplify. For example, $$ \sqrt{-20}=i \sqrt{20}=i \sqrt{4} \sqrt{5}=2 i \sqrt{5} $$ $$\sqrt{-49}$$

Step-by-Step Solution

Answer

The simplified form of \(\sqrt{-49}\) is \(7i\).

Step 1: Identify the Negative Sign

Recognize that the expression involves taking the square root of a negative number, specifically \(-49\). The negative sign indicates that an imaginary unit \(i\), where \(i = \sqrt{-1}\), will be used in the simplification.

Step 2: Rewrite as the Product

Express \( oot{-49}\) as \( oot{-1} imes oot{49}\), separating it into the product of the square root of \(-1\) and the square root of \(49\). This becomes \(i imes oot{49}\).

Step 3: Simplify Square Root of Positive Number

Calculate \( oot{49}\), which is the square root of a positive number. Since \(49\) is a perfect square, \( oot{49} = 7\).

Step 4: Combine Results with the Imaginary Unit

Combine the results from Steps 2 and 3: \(i imes 7 = 7i\). Thus, the simplified form of \( oot{-49}\) using \(i\) is \((7i)\).

Key Concepts

Complex Numbers

Complex numbers are numbers that combine both real and imaginary parts. A complex number is typically written in the form \(a + bi\), where \(a\) represents the real part, and \(b\) is the coefficient of the imaginary part \(i\). The letter \(i\) is used to denote the imaginary unit, defined as the square root of -1, or \(i = \sqrt{-1}\).

In our example, the expression \(\sqrt{-49}\) is simplified to \(7i\). This is a complex number where the real part is 0, and the imaginary part is 7 times \(i\).

• Real part: 0
• Imaginary part: 7

Overall, complex numbers allow us to deal with situations that involve the square roots of negative numbers, extending our number system in a very useful way for many fields such as engineering and physics.

Square Roots

A square root of a number is a value that, when multiplied by itself, gives the original number. For positive numbers, this is straightforward. For example, the square root of 49 is 7 because \(7 \times 7 = 49\).

However, when dealing with negative numbers, square roots require special treatment through imaginary numbers. The expression \(\sqrt{-49}\) can be rewritten using the imaginary unit \(i\), which corresponds to \(\sqrt{-1}\).

• Separate the expression: \(\sqrt{-49} = \sqrt{-1} \times \sqrt{49}\).
• Recognize \(\sqrt{-1}\) as \(i\).
• Calculate \(\sqrt{49}\) as 7, since 49 is a perfect square.

Putting it all together gives \(i \times 7 = 7i\), a manageable way to represent the square root of a negative number.

Simplification

Simplification involves transforming a mathematical expression into its most concise and easily understood form. It's about reducing complexity while maintaining equivalence to the original expression.

In dealing with expressions like \(\sqrt{-49}\), the process of simplification helps in representing and manipulating the expression using complex numbers.

• Identify negative square roots and rewrite them using \(i\).
• Separate the imaginary unit \(i\) from the positive square root part.
• Simplify any perfect squares for easier computation.
• Combine the real and imaginary parts into a single expression.

This step-by-step simplification results in a clean and precise representation, \(7i\) in this case, which is far more useful for computation and analysis.