Problem 47 - Precalculus Mathematics for Calculus

Question

Evaluate the radical expression and express 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 Radical Expression

The given expression is \( \sqrt{-25} \). Observing the negative sign under the square root tells us that we have an imaginary number.

Step 2: Rewrite using Imaginary Unit

Rewrite the square root with the imaginary unit \( i \), where \( i^2 = -1 \). Thus, \( \sqrt{-25} \) can be rewritten as \( \sqrt{25} \cdot \sqrt{-1} = \sqrt{25} \cdot i \).

Step 3: Simplify the Expression

Now simplify \( \sqrt{25} \) which equals 5. Therefore, \( \sqrt{25} \cdot i = 5i \).

Step 4: Express in the Form \(a + bi\)

In the form \( a + bi \), the real part \( a \) is 0, and the imaginary part \( b \) is 5. Therefore, the expression is \( 0 + 5i \).

Key Concepts

Imaginary Unit

Imaginary numbers might sound complex, but they simply involve the imaginary unit, denoted as \(i\). The fundamental property of \(i\) is \(i^2 = -1\). This quirky little unit allows us to handle square roots of negative numbers within math. In essence, it lets us explore a whole new dimension of numbers called complex numbers.
While real numbers are like the reliable old friends of the number world, imaginary numbers give us one more friend who opens up different possibilities. Introducing \(i\) into math helps solve equations that don’t fit into the world of real numbers. Whenever you see \(\sqrt{-x}\), think of \(i\). It's like a key that unlocks these seemingly unsolvable roots of negative numbers.

Square Root

The square root is a familiar operation in math where you find a number that, when multiplied by itself, gives you the original value. For example, the square root of 9 is 3 because \(3 \times 3 = 9\). But what happens when we see a negative number under that square root? That's where things get interesting!
Mathematicians have devised methods to deal with these by using the imaginary unit, \(i\). Instead of leaving \(\sqrt{-x}\) hanging as an unsolved problem, we can express it through multiplication involving \(i\). Thus, \(\sqrt{-25} = \sqrt{25} \cdot \sqrt{-1}\) and further simplifies to \(5i\). This method allows us to evaluate expressions that traditional methods would struggle to find solutions for.

• The square root of a negative number always includes \(i\).
• This process opens our understanding to complex numbers.

Radical Expression

A radical expression refers to any expression that includes a root symbol, such as a square root. In the given exercise, \(\sqrt{-25}\) is our radical expression. Recognizing when a radical involves an imaginary component is crucial in simplifying and solving these expressions.
This begins by identifying if the number under the root is negative. If it is, like \(-25\) here, you replace the radical with \(\sqrt{positive\,component} \cdot \sqrt{-1}\). This translates into simplifying the numeric part \(\sqrt{25}\) and attaching \(i\), yielding \(5i\) in this case. Such transformations are essential for expressing complex numbers seamlessly.
Breaking down the radicals into manageable parts helps simplify and understand expressions involving imaginary numbers. By following these transformations, you can tackle different radical expressions with ease.