Problem 43 - Algebra and Trigonometry

Question

Evaluate the expression and write the result in the form \(a+b i .\) $$ \sqrt{-25} $$

Step-by-Step Solution

Answer

The expression evaluates to \(0 + 5i\).

Step 1: Recognize the Imaginary Unit

Recall that the square root of a negative number involves the imaginary unit. The imaginary unit is denoted by the symbol \( i \), where \( i^2 = -1 \). This means \( \sqrt{-1} = i \).

Step 2: Break Down the Negative Square Root

Given the expression \( \sqrt{-25} \), rewrite it as \( \sqrt{25} \cdot \sqrt{-1} \). This is possible because the property \( \sqrt{a \cdot b} = \sqrt{a} \cdot \sqrt{b} \) holds true.

Step 3: Compute the Square Roots

Now, find the square roots: \( \sqrt{25} = 5 \) and \( \sqrt{-1} = i \).

Step 4: Combine the Results

Multiply the results from Step 3 to get \( 5 \times i = 5i \).

Step 5: Express in the Form \(a+bi\)

Write the final expression in the form \(a+bi\). Here, since there is no real part, it's \( 0 + 5i \). Therefore, it is expressed as \( 5i \).

Key Concepts

Imaginary Unit

The concept of the imaginary unit is foundational when dealing with complex numbers and square roots of negative numbers. The imaginary unit is denoted by the letter \( i \), which is defined as the square root of \( -1 \). In mathematical terms, this means \( i^2 = -1 \). This definition allows us to work with numbers that would otherwise not exist within the realm of real numbers.

• The imaginary unit forms the basis of complex numbers, enabling calculations involving negative square roots.
• When you see \( \sqrt{-1} \), it simplifies to \( i \).

Understanding the imaginary unit sets the stage for diving deeper into complex numbers and their applications in mathematical expressions.

Square Root of Negative Numbers

The square root of negative numbers can seem puzzling at first. Normally, square roots of positive numbers are straightforward. For example, \( \sqrt{25} = 5 \) because \( 5^2 = 25 \). But what about \( \sqrt{-25} \)?In order to solve \( \sqrt{-25} \), you can separate the expression into two parts: \( \sqrt{25} \times \sqrt{-1} \). This uses the property \( \sqrt{a \cdot b} = \sqrt{a} \cdot \sqrt{b} \).

• Calculate \( \sqrt{25} \), which equals \( 5 \).
• Then, \( \sqrt{-1} \) is known as \( i \), the imaginary unit.

Putting them together, \( \sqrt{-25} \) becomes \( 5 \times i = 5i \). The expression is thereby simplified using the imaginary unit.

Complex Number Form a+bi

Complex numbers come in the form \( a + bi \). Here:- \( a \) represents the real part.- \( b \) is the coefficient of the imaginary unit, \( i \).When working with complex numbers, such as \( 5i \), you are effectively looking at \( 0 + 5i \), which means:

• The real part \( a \) is 0.
• The imaginary part \( b \), placed in front of \( i \), is 5.

The form \( a+bi \) is essential for expressing any complex number, helping organize both real and imaginary components. In our problem, the expression \( 5i \) is listed in this form with \( a = 0 \) and \( b = 5 \), clearly highlighting that the number has no real part but consists solely of an imaginary component.